Comprehensive Guide to Logarithms

In this video we're going to... go through logarithms. This is going to be a complete guide to working with logarithms. We're going to talk about how to rewrite logarithms from exponential form to logarithmic form and from logarithmic form to exponential form. We're going to talk about expanding and condensing logarithms. We're going to talk about the change of base form though. We're going to talk about how to solve equations involving logs. So this is going to be like a comprehensive video going into different aspects of working with logarithms. Let's dive in. I'm going to have a lot of different examples. You can practice some of these on your own to get some practice. practice. So we'll do multiple types of each kind. So let's dive into this first set of problems. We're going to talk about how to rewrite logarithms into the exponential form. So the first thing you want to know is this form here. This is the log base B of X equals N. And when we rewrite it, see this is log base B. This is an exponential function with base B. And notice that here is the power and X is like your answer. So what some students do when they're rewriting is they say, well, I know this is my log base b. I know this is my exponential function base b. All I have to do is interchange these two quantities. That's one way to do it. I'll show you another way that I like that's a little bit more intuitive. So let's take a look at a couple examples. So for number one, log base 4 of 64 equals 3. Because logarithms and exponential functions, they're inverses of each other, they're kind of like squaring and square rooting or like multiple. multiplying and dividing, they're like, they undo each other, they're inverses. What I can do is I can exponentiate, I can raise both sides using this base 4. So see how this is an exponential function with base 4 and a logarithmic function of base 4? Those undo one another, they're inverses, and now you can see it's an exponential form. Now I could rewrite this a little bit if I want, I could say 4 to the third power equals 64, that's 4 times 4 times 4, three times, and you've got it. So this is kind of a... of a little bit more of an intuitive way, kind of like when you had a problem like this, like x squared equals 16, and you say, oh, I'm going to just take the square root of both sides, see the square and the square root, those like are inverses, they undo each other. Of course, the first time you saw that symbol, you were probably like, you know, what is that, right? It's like, and it just means, you know, the inverse or the opposite, right? So let's go to number two now, log base 5 of 1 is 0. So if I was going to rewrite this, again, you could use this technique that I mentioned here, where you say, oh I recognize this is base 5, all I have to do is interchange these two numbers here, 0 equals 1. Or you can do the method that I kind of like which is to exponentiate. I can raise both sides using that base 5. These undo each other and you can see 5 to the 0 power equals 1. Same thing. Let's go to number three. See if you can do this one. Now this one's a little bit different. Notice how it says log and there's no base. So what does that mean when there's no base? Well this is called a common log. and this is understood to be base 10. Now you can write that base 10 in there just to kind of remind yourself and then again you can use the technique that I mentioned here where you keep the base the same base 10 you interchange these two numbers Or you can do the technique that I prefer, which is to exponentiate both sides using base 10. So there's an exponential function. There's a logarithmic function, kind of like squaring and square rooting. They're opposites. And 10 squared is 100. The negative, you take the reciprocal. That's how we're getting 1. So now it's in the exponential form. Now you're probably saying Mario, you know, why are we even doing this? You know, why are we switching forms? Well, then we're gonna go through switching from logs to exponential form then exponential to logarithmic form But then we're going to talk about how to evaluate logs, which means how do you find the value, and that's going to come in handy when we get to that section of this video. So see if you can do number four and five on your own. Go ahead and pause the video just to test yourself, see if you're grasping this. Now again if I was going to do this I would just exponentiate both sides using this base 125. The exponential function and the logarithmic function are inverses and you can see I have 125 to the 1 3rd power equals 5. which makes sense because the 1 3rd power is like the cube root and we know the cube root of 125 is equal to 5. And for the last example what did you get for this one? Well again you can use this technique I was mentioning here where you can keep the base the same. and just interchange these two quantities. So 2 to the 5th power equals 32, which makes sense because 2 times 2 times 2 times 2, 5 times is 32. Or you can use my preferred technique, just because it's a little bit more intuitive, you can exponentiate both sides using base 2. These are inverses, and this makes sense. 2 to the 5th is 32. So let's now talk about going the other direction from exponential function back to the logarithmic form. Okay for this next group of problems we're going to switch from the exponential form back to the logarithmic form. So I'll show you my preferred method here first. So what I would do is I'd say this is an exponential function with base 3. So I'm going to do the inverse which is to take the log base 3 of the left side, log base 3 of the right side to keep it balanced right, and then the log base 3 and the exponential function base 3, as long as the bases are the same these are inverses they cancel one another out. So now we have log base 3 of 81 equals 4. Or if you want you can flip this whole equation and write it like this. Log base 3 of 81 equals 4. Now that's in logarithmic form. Now if you want to check your work you can always rewrite it back into the exponential form. You can say oh this is log base 3. How about if I exponentiate both sides using base 3? These would cancel one another out and 3 to the fourth equals 81 which was our original problem. And you might want to do this actually keep going back and forth say okay now it's an exponential form let me go back to logarithmic form and just keep switching just so you feel comfortable changing forms okay. They're equal to either equivalent it's just written in a different way right. So for number seven let's do this one together so we've got seven to the negative second power equals one over forty nine we know that's true because seven squared is 49 the negative you take the reciprocal okay but all we're doing is we're going to rewrite it in the log form by taking the log base 7 of both sides of the equation. Keep it balanced. And the reason this works is see log base 7, exponential function base 7, those undo one another and now you can see that it's in the logarithmic form. Now you might want to flip it just so that the logs over here on the left. Doesn't really matter. It's a you know an equation but just to see it from a different perspective. That's now in log form. So if you want you're getting the hang of this try it number 8, 9, and 10 on your own. And go ahead and pause the video and see if we can match what we're getting here. So 2 to the 0 power equals 1. Of course we know that whenever you raise something to the 0 power that's equal to 1. But in log form let's go ahead and take the log base 2. So you want these guys to match. Whatever you do to the left side you do to that to the right side. You know that from algebra 1 right? And so these undo each other and now you can see log base 2 of 1 equals 0. I might just want to rewrite it a little bit. just so it reads a little bit nicer. Log base 2 of 1 equals 0. And one thing that you'll notice is that whenever you take the log of 1, no matter what the base is, that's always going to equal 0. So that's kind of a nice thing to remember. So for number 9, what did you get for number 9? Well let's see. 16 to the 1 half power equals 4. Well the 1 half remember is like the square root of 16 which is 4. But all we're doing is saying let's write it in the log form. Let's take the log base 16 of both sides. And now you can see it's in the logarithmic form. I'm just going to flip the equation log base 16 of 4 equals 1 half. And now it's in our log form. Now, again, you can do this another way. You could say, let's say, for example, number 10. What did you get for this one? I could say, well, here's my base, right? So I could say there's log base 3. And remember how we said we switched the 2 and the 9? So this would be log base 3 of 9 equals 2. If you like that technique, you know, go with that. Some people, that's more comfortable. They say, oh, I know my base is 3, so log base 3. I just switch the exponent and the answer, and now it's in the logarithmic form. If you don't like that method, you can do my method, which is just kind of like to do the inverse or the opposite. I would take the log base 3 of both sides, okay? And then as long as these bases are the same, the log base 3 and the exponential function base 3, log base 3 of 9 is equal to 2, you're going to get the exact same thing. So now let's go ahead and talk about... Evaluating logs. Okay, the next 10 problems we're going to talk about how to evaluate logs. Evaluate just means find the value, find out what the logarithm is equal to, right? So let's start with number 11 here. Log base 4 of 16 equals blank. Now some students they like to put a variable there, say for example like x. And then what we're going to do is we're going to use the skills that we've already mastered, which is to rewrite log equations into exponential form. So what I'm going to do is I'm going to exponentiate or raise both sides using this base 4. This way the exponential function and the logarithmic function, they undo one another or cancel each other out. So now I just have to answer this question. 4 to what power is 16? Well, you can see that's going to be 2. So this whole thing, the value of this logarithm is equal to 2. And you got it. Let's try number 12. So this one, log base 3 of 1 27th equals. Now some students... like to put a little question mark. So they'll say okay how can I solve this? Now you can do my method which I've been showing you is just to exponentiate both sides. These are inverses. Three to what power is 1 27th? Well I know 3 cubed to the third power that's 27 but because it's the reciprocal this is actually going to be a negative exponent. It's going to be negative 3. For number 13 let's just go with making it equal to a variable x for now. And you can use the technique we were talking about here where we say, okay, this is log base 5. I'm just going to switch the x and the 1 so you're just switching the exponent and the answer and now we just have to answer this question. 5 to what power is 1? That's 0. And remember how we talked about whenever you take the log of 1 no matter what the base is it's 0? And this is this is y right? So for number 14 let's look at this one. Log base 9 of 3 equals what? What's the value of this log? Let's exponentiate both sides using base 9. You always want to make sure these bases match. Those undo each other. 9 to what power is 3? Well 9 to the 1 half power is 3 because 1 half is like the square root. So if it was the 1 third that'd be the cube root or 1 fourth that's the fourth root. So the square root of 9 is 3 that that matches. And for number 15 now remember log of a thousand there's not a base here. So what's the base when there's no base? Base 10 right? That's a common logarithm. It's a Commonly used on your calculator you'll see log, L-O-G, which some people ask me sometimes, Mario what's this 10G? That's a L, that's a L-O-G, right? But now you just say, okay common log, so this is going to be exponentiate both sides. So these are inverses. 10 to what power is a thousand? Well 10 times 10 times 10, three times is a thousand, so the answer is three. Let's do five more examples and maybe you can practice these ones on your own. Now before you dive into some of these problems on your own, I want to explain a little something about this Ln. So when you see Ln, this is what's called the natural log. And Ln, you can write it like this if you want, log base e. So natural log, it means it's log base e. But as a shorthand, we just write it as Ln or natural log. So again, sometimes it helps to put an e here as a base, just to remind us that it's base e. Just like when we deal with log, we don't see the base. That's understood to be a common log or base 10. So let's do this one. Natural log of 1 is equal to blank. We know this is at log base e, so what I'm going to do is I'm going to exponentiate both sides using base e. Those are inverses. e to what power is 1? Well, e to the 0 is 1, so this is just going to be equal to 0. So if you want, you can try some of these other ones, 17, 18, 19, and 20, if you want to pause the video. For 17, again, this is... natural log which is understood to be log base e so I'll put the base there. Notice the base is usually a little bit smaller and it's lower. Okay so that's how you know it's the base. And one thing that's interesting about this particular problem is see how this is log base e and an exponential function base e. So if you can get these to match it's just like we've been talking about using inverses this would just come out to four. Now if you don't like that method because some students they just like to kind of do it the same way each time which is completely fine. You just say well here's log base e, let me exponentiate both sides using base e. Those are inverses. e to what power is e to the fourth? Well if the bases are the same then the exponents have to be the same and x would equal 4. So you're going to get the same answer. But sometimes if you can catch it early enough and say oh the bases match, one's exponential, one's logarithmic, they undo each other, it's just a little quicker. So what did you get for 18, 19, and 20? Well for 18 I notice there's not a base here so this is understood to be base 10. So I'm going to say exponentiate both sides. Those are inverses. 10 to what power is the square root of 10? Well we know the one half power is the square root so 10 to the one half is the square root of 10. For number 19 what did you get on this one? This one I'm going to do it a little bit differently just to show you. 128 can I get that in terms of base 2? Well yes because I know that 128 is 2 to the 7th power. Let's see 2, 4, 8, 16, 32, 164, 128. And so now I can see I've got log base 2 and exponential function base 2. And so you can see we're just getting 7. Again you don't have to do it that way. You could just exponentiate both sides. 2 to what power is 128? That'll be 7. You'll get the same answer. And then for number 20, what did you get for this one? Again, we'll just do the method we've been using, which is to do the inverse or to exponentiate. 1 5th to what power is 25? Well, 1 5th to the negative 2 power, because we know 1 5th squared is 1 25th, but the negative we take the reciprocal and that's how we're getting 25. Now before you dive into some of these problems on your own, I want to explain a little something about this Ln. So when you see Ln, this is what's called the natural log. And Ln, you can write it like this if you want, log base e. So natural log, it means it's log base e, but as a shorthand we just write it as Ln or natural log. So again, sometimes it helps to put an e here as a base, just to remind us that it's base e. Just like when we deal with log we don't see the base that's understood to be a common log or base 10. So let's do this one. Natural log of 1 is equal to blank. We know this is a log base e so what I'm going to do is I'm going to exponentiate both sides using base e. Those are inverses. e to what power is 1? Well e to the 0 is 1 so this is just going to be equal to 0. So if you want you can try some of these other ones 17, 18, 19, and 20 if you want to pause the video. For 17, again, this is natural log, which is understood to be log base e, so I'll put the base there. Notice the base is usually a little bit smaller and it's lower. Okay, so that's how you know it's the base. And one thing that's interesting about this particular problem is, see how this is log base e and an exponential function base e? So if you can get these to match, it's just like we've been talking about using inverses, this would just come out to 4. Now if you don't like that method, because some students, they just like to kind of do it the same way. each time, which is completely fine. You just say, well here's log base e, let me exponentiate both sides using base e. Those are inverses. e to what power is e to the fourth? Well if the bases are the same then the exponents have to be the same and x would equal four. So you're going to get the same answer, but sometimes if you can catch it early enough and say, oh the bases match, one's exponential, one's logarithmic, they undo each other, it's just a little quicker. So what did you get for 18, 19, and 20? Well for 18 I notice there's not a base here so this is understood to be base 10. So I'm going to say exponentiate both sides. Those are inverses. 10 to what power is the square root of 10? Well we know the one half power is the square root so 10 to the one half is the square root of 10. For number 19 what did you get on this one? This one I'm going to do it a little bit differently just to show you. 128 can I get that in terms of base 2? Well yes because I know that 128 is 2 to the 7th power. Let's see 2, 4, 8, 16, 32, 164, 128. And so now I can see I've got log base 2 and exponential function base 2. And so you can see we're just getting 7. Again, you don't have to do it that way. You could just exponentiate both sides. 2 to what power is 128? That'll be 7. You'll get the same answer. And then for number 20, what did you get for this one? Again, we'll just do the method we've been using which is to do the inverse or to exponentiate. 1 5th to what power is 25? Well, 1 5th to the negative 2 power. Because we know 1 5th squared is 1 25th, but the negative we take the reciprocal and that's how we're getting 25. Okay, for these next four examples we're going to talk about how to graph logarithms. So let's look at this first example, number 21. y equals log. of x. Now there's not a base here so we know this is a common log or base 10. Let's rewrite it into the exponential form. So the way we do that remember is we exponentiate both sides okay using the same base these are inverses and so now you can see we have 10 to the y equals x. Now we know that y equals 10 to the x okay looks something like this it's an exponential growth graph. Okay, because see the base is greater than one so it's going to be an exponential growth. But what's the difference between this equation and this equation? All we really did was interchange or switch the the x and the y and that's how you find the inverse. You're interchanging the x and the y. When you interchange the x and the y, graphically what that does is it reflects the graph over the line y equals x, this 45 degree line, and you get a graph that looks something like this. So let's go ahead and make a table. I'll show you how I normally do this. I make a table. But what I do is instead of plugging in for x, I'm going to plug in for y since that's in the exponent position. So let's plug in some easy numbers. 10 to the negative 1. Remember the negative exponent you take the reciprocal. So that's 1 tenth. 10 to the 0. Anything to the 0 power is 1. And then 10 to the first. Anything to the first power is itself or 10. If I plot these points now, 1 tenth down 1 is going to put us right about here. 1, 0 is going to be right here on the x-axis and 10, 1 is going to be right 10 up 1 like this. So you can see our graph is going like... Something like this. And notice it's getting closer and closer to the y-axis. So we have a vertical asymptote. This is the asymptote x equals 0, this vertical line. And what happens is the domain is going to be greater than 0. Or if you're using interval notation, you can say from 0 to infinity, like that. And then the range is going to be all real numbers. Because even though this is going to the right, It looks like it's almost horizontal, but it's gradually going up and this is going all the way down to negative infinity So we would say the range is all real numbers or negative infinity to positive infinity. So that's a good technique You can basically rewrite it into the exponential form Substitute in for y instead of x make a table and plot your point So that's a good way to do it. But looks like at number 22. This one's involving a little bit of uh, Transformations now, what do you think this one does to the graph? And what do you think this 3 does to the graph? Well if you've been studying transformations, you know that the one group of the x has like the opposite effect. So the minus 1 is actually going to shift right 1, see positive 1, and the plus 3 has the same effect, it's going to go up 3. So what we can do if we want is think of this right 1 up 3 as like our new starting point, kind of like our new origin. I could even draw in like a new y-axis and like a new x-axis. We'll think of that as our starting point. I'm going to ignore the 1 and 3 for a moment and I'm going to rewrite this in exponential form. So y equals log base 2 of x. I'm going to exponentiate both sides using base 2. And so now you can see we have this equation 2 to the y equals x. Let's make a table, plug in some small numbers, okay, and let's see what we get. So 2 to the negative 1 is 1 half. 2 to the 0 is 1. 2 to the first is 2. 2 to the second is 4. 2 to the third is 8, etc. Notice I'm plugging in for the y. That makes it a little bit easier to evaluate. So now if we plot these points, but instead of from our original origin, let's use our shifted origin. So I'm going to go from here, right a half, down 1, which is right about here. 1, 0. So from here, right 1, up 0. Then right 2, up 1. Right about here, and then let's say right 4 up 2. So this would be 2, 3, 4, up 2, right about here. Okay, so there's our graph, and you can see the domain now is going to be greater than 1. So you can say x is greater than 1 but not equal to 1, or if you're using the interval notation you can say from 1 to infinity. And then the range, the y values, you can see it's going down. Gradually up forever and ever so we can say the range is all real numbers or you could say negative infinity to positive infinity If you're doing the interval notation and then the equation of this vertical asymptote right here remember vertical lines are x equals lines This would be x equals 1 Okay, now another thing you can do too is you can't take the log of 0 or a negative quantity So whatever is in parentheses here, whatever you're taking the log of it has to be positive meaning this quantity has to be greater than 0 So you could solve this inequality, add 1 to both sides. Okay, as long as x is greater than 1, that's my domain. Or if you set it equal to 0, like x minus 1 equals 0, and solve, that gives you the equation of your asymptote x equals 1. Now this is just one way to do it using the transformations. If you don't like that method, what you can do is just take this entire equation and rewrite it. okay, an exponential form. And the way I would do that is I would work from the outside in towards this x. So I would subtract three from both sides. So this will be y minus three equals log base two of x minus one, okay. Then I would exponentiate both sides using base two, okay. So now I just have to add one to the other side. So I get two to the y minus three plus one equals x. Now it's an exponential form. You can plug in values for y. and you're going to get the output your x. You can make a table. Just make sure if you use the entire equation that you graph from the original origin, the zero zero point. That's one way students sometimes like to do it. If it's a simple transformation like this, I just like to kind of look at my parent function and graph it from the shifted origin. Let's look at two more examples. Maybe you can practice these on your own to get some more experience. Okay, if you want to try some of these on your own, try 23. This is a good one to practice. Y equals log base one-third. of x plus 2 minus 1. How would you graph that one? Well if I was going to do it I would make use of the transformations. I'd say well this number grouped with the x is going to have the opposite effect. It's going to shift left 2. This number that's not grouped with the x that's going to be a vertical shift down 1. So if I go left 2 down 1 I can think of this as my new origin. I'm going to draw in like a new x-axis and a new y-axis. So I'm going to think of this as my starting point. Then I'm going to focus in on the parent function. The parent function is just like what's left here, log base one-third of x. And I'm going to rewrite this in exponential form by exponentiating or raising both sides using base one-third. So now I can say, one-third to the y power equals x. So if y is negative one, one-third to the negative one is three, because the negative exponent you take the reciprocal. One-third to the zero is one, because anything to zero power is one. And then 1 third to the first would be itself 1 third. So if we plot these points now from this new starting point, let's see, we've got right 3, down 1, right 1, up 0, right 1 third, up 1. You can see this graph looks like this. Now that might look a little strange to you. You might be saying, Mario, I thought you said the law graphs look like this, right? So what happened? Well remember where we were dealing with our parent function y equals log base one-third of x, right? See if we were to graph y equals one-third to the x power notice that this is a base that's less than one between zero and one that's an exponential decay function. Remember exponential decay functions they go like this down to the right they approach that horizontal asymptote of the x-axis so they look like that. And remember when you find the inverse, what are you doing? You're finding the reflection of that graph over that line y equals x. So if I was to take that graph and reflect it, it's going to look something like this now, where it's like getting closer, it's going to go kind of like that. Okay, and that's what we have here. It's just that this graph has been shifted left 2 and down 1. So if you want to focus on the asymptote here, you'd say x equals negative 2. Sometimes they ask you for the equation of the asymptote. It'll ask you for the domain so you can say x has to be greater than negative 2 or you could say an interval notation negative 2 to negative infinity. And then the range is going down and up forever and ever so you could say the range is all real numbers or you could say from negative infinity to positive infinity for the range. Okay so good. Let's do number 24. Now this one you might need a little bit of help on if you haven't seen this before. y equals natural log of x. Remember the natural log this is like log base e. So I'll just put the either to remind us. Let's rewrite it in exponential form by Exponentiating both sides. So we're really trying to graph e to the y equals x. So let's plug in some numbers for y like negative 1 0 and 1. e to the negative 1. Now we didn't talk about this at the very beginning, but this is the natural base e. Right, and e is one of those numbers kind of like pi. You know how pi is like a non-repeating, non-terminating, like 3.14159, like that. E is about approximately 2.7. So you might want to memorize that. About 2.7. So if we do e to the first, that of course is going to be itself, 2.7. If we do e to the zero, anything to zero power is one. If we do e to the negative one, that's like one over e. And 2.7 is almost three, okay? So it's about one third. I'll just say approximately. 1 third. So now when we plot these points, let's do that. 1 third, down 1, 1, 0 means we're going to write 1 up 0, and then 2.7, 1, 2, almost a 3, not quite, up 1. So now if we graph this, there's our graph. So it's approaching this y-axis, which is the asymptote x equals 0. The domain is going to be x is greater than zero or from zero to infinity if you're using interval notation and the range is going to be all real numbers or negative infinity to positive infinity. But again this is helpful to rewrite it in the exponential form. Okay now we're going to talk about expanding and condensing logs. These are called the properties of logarithms. You probably remember when you learned about the properties of exponents. Let's go through it real quick. When you multiply and you have the same base What do you do to the exponents? You add them, right? Similarly, when you work with logs, if you have the log of a product, see x times y, you write it as the sum log base b of x plus log base b of y. So just like when you multiply you add, when you multiply you add. Just want to make sure you keep the bases the same. This is called the product property of logs. When you go from the left side, To the right side we call that expanding. It's getting bigger. If you go from the right side to the left side we call that condensing. It's getting smaller. You're combining it into just one log. Now let's look at the quotient property. When you divide with exponents and you have the same base, what do you do to the exponents? You subtract them. With logs, when you have log base b of x divided by y, what do you do to the logs? You subtract them. You take log of the numerator minus log of the denominator. This is called the quotient property of logs. And then here, when you have a power to a power with exponents, like an exponent raised to an exponent, what do you do to the exponents? You multiply them. With logs, when you have log base b of x to the nth power, what do you do with this power? You can bring it down in front of the log and multiply it by that n. So this would be n times log base b of x. It's also reversible, so you could take this coefficient and bring it up as a power. That would be condensing. When you bring it down, that's called expanding. So this is called the power property of logs. So we're going to go through some examples together and you can practice some on your own if you want to pause the video. But I'll see if I can show you how this works. Let's do this first example log f u n. Log of fun right? Fun with logarithms right? Well see here's the thing. They don't tell us the base. That's understood to be base 10. We can see that f u and n are right next to each other. That means they're multiplied together just like this one here. So we can write it as a sum. logarithms. So this is going to be equal to log of f plus log of u plus log of n. That's expanded. Now if you if they gave us this we could condense it back into log of f times u times n. So we're combining it into one log. But in this case we're going to focus on expanding first. Okay let's go to number 26. This one we have log of wx divided by y. Now you can see this fraction bar is like a division sign. That's like this one, the quotient property. When we divide we subtract. So we could say log of wx minus log of y. But notice the w and the x are right next to each other. That means they're multiplied together. That's this product property of logs. We can write it as a sum. So when you multiply you write it as a sum. So log of w plus log of x minus log of y and now it's fully expanded. Let's go to number 27. This one log base 4 of L divided by MN. So this one we've got a different base. We have base 4. So I'm going to say log base 4 of L minus log base 4 of MN. See remember when you divide you subtract. But notice that M and N are multiplied together. So what I'm going to do is I'm going to write this as a... Product, okay, you write as the sum of logs. So this would be log base 4 of m plus log base 4 of n. Now here's where students go a little bit off the tracks. See this minus sign? This whole thing is being subtracted. That means this whole thing here is a group. So we're just expanding this, but the whole thing is being subtracted. On the next step I'm going to distribute that negative into the parentheses and this is going to be log base 4 of L minus log base 4 of m. minus log base 4 of n fully expanded. Now when we get to the condensing and I'll talk about that a little bit more then but whatever logs are being subtracted those arguments, okay these are that what are called the arguments not like getting into a fight just like they just call them the arguments, the ones that are being subtracted those logs those quantities go in the denominator. The ones that are positive or added those arguments are going to go in the numerator. So that's helpful to remember. After doing this a while, you'll be able to jump right from here to here. You'll say, oh, L, that's a positive log. Oh, these ones are going to be negative or they're going to be subtracted because they're in the denominator. So you can kind of skip steps after a while. But in the beginning, I would just take it step by step. Now for number 28, this one, notice we have an exponent here. This is our power property of logs. We're going to bring that power down in front of the log and multiply it. So we're going to say 3 times the natural log of x. That's fully expanded. Now if we were going to condense we would bring the 3 back up as a power. We get back to the original and that's considered condensing, like it's more condensed. Let's look at four more of these expanding examples and then we'll get into the condensing. Okay for these next four examples if you want to try them on your own go ahead and pause the video. See if you can do them. See how far you can get. Let's dive in number 29. How would we expand this? Well we can see this is log base 5. We can see that the w squared and the z to the fifth are next to each other. That means they're multiplied together. So we're going to use our product property. So if I was going to do this, I would say this is log base 5 of w squared plus log base 5 of z to the fifth. Now what do we do when we have an exponent? That's our power property. We can bring that power, that exponent, down in front. and multiply it by the log. So this is just going to end up being 2 log base 5 of w plus 5 log base 5 of z. And that's fully expanded. Now if you want to check your work you can reverse it, you can condense it. I could bring the 2 back up as a power and the 5 back up as a power and if I'm adding I multiply those arguments together and I get back the original. So you can check your work on these. For number 30 how would you do this one? Log base 3 of the fourth root of x times y. Now you can do this a couple different ways. One way to do this is the fourth root is really like the 1 fourth power. So this is like xy to the 1 fourth. Now you can bring the 1 fourth down in front, right? So let's do that. That's the power property. 1 fourth of log base 3 of x times y. When they're multiplying, remember that's this product property, you can write it as a sum. So this is going to be log base 3 of x plus log base 3 of y. But that whole thing is multiplied by 1 fourth. Sometimes people just multiply the first one. They forget about the second one. So you're going to actually want to distribute that 1 fourth into the parentheses. So it's going to be 1 fourth log base 3 of x plus 1 fourth log base 3 of y. That's fully expanded. And again, you can check your work by reversing. those steps and condensing it. For number 31 we have log base 2 of cat divided by dog. Or you could say the product of cat divided by the product of dog. So here what I would do, this one I'm going to be a little bit shortcutty about this. Whatever is in the numerator, remember we said that those are going to be positive logs and everything that's in the denominator those are going to be subtracted or negative logs. So this is going to be log base 2 of c plus log base 2 of a plus log base 2 of t minus log base 2 of d minus log base 2 of o which kind of looks like a zero you can't take the log of zero but just know that that's a an o minus log base 2 of g so see how i did that so now you might be saying mario why does that work well one way to think about it is when you subtract subtraction is like adding the opposite right And this minus sign is kind of like a negative 1, right? And this negative 1 you can bring up as a power, that's our power property, right? So this could be like g to the negative 1. And the negative 1 tells us to take the reciprocal, so that would be like 1 over g. And when we add, we multiply, but if you multiply by 1 over g, that's going to bring the variable down to the denominator. So different ways to look at this when you're doing these problems. Let's do 32. This one... We've got a lot going on. Okay, so what I'm going to do here is I'm going to rewrite this a little bit. I'm going to say log base 7 of x to the 2 thirds. See how the numerator is the power and the denominator is the root? Times y to the 1 third, okay, all over z to the 4th. Okay, so now that's a little bit rewritten. So now remember when we multiply, we add, and when we divide, we subtract. So this is going to be log base 7 of x to the 2 thirds. plus, because when you multiply you add, log base 7 of y to the 1 third, minus when you divide log base 7 of z to the 4th. Now what we're going to do is we're going to use our power property. We're going to bring all these exponents down in front, and we have it fully expanded. So this is going to be 2 thirds log base 7 of x plus 1 third log base 7 of y minus 4. log base 7 of z and now you've got it fully expanded. Okay see if you can do these condensing ones on your own and we'll do four of these easier ones. We'll have a couple harder ones for the last two. So what would you do for 33? Here you're adding. How could you condense this into a single logarithm? Well when you add what we're going to do is we're going to go from the right side to the left side. We're going to write it as a product. So this is actually going to be 5 times 3. Or we could just say log of 15. And for the next one, number 34, we're subtracting. What do you do when you subtract? You divide. So this is really going to be log of 3 over y. Now some students make a mistake. They'll say log of 3 divided by log of y. You just want to condense it into one log and like that. So you want to make sure it's just one logarithm. For 35, this one's a little bit more challenging. What I would do is I would bring the... coefficients up as powers first, that's our power property, we're just bringing that up as an exponent, and then when you subtract, you divide. So this is going to be log of x squared minus log of y cubed but when you subtract you're going to divide the argument so this is going to be log of x squared over y cubed. Again just one logarithm. For number 36 same thing here we're going to bring up the coefficients as powers. Remember the one half power is the square root so I'm going to really write this as a natural log of the square root of w. minus natural log of y squared and then this is a natural log of x to the fourth. Remember when you subtract you divide, when you add you multiply. The way I like to think of it is all the positive logs, the arguments go in the numerator, all the ones you're subtracting are negative, those arguments go in the denominator. So this would be natural log of x to the fourth times the square root of w divided by y squared. Now another mistake that students sometimes make is they'll take this fraction bar and extend it so that it's the nat even the natural log being divided. You don't want to divide the natural log just the arguments over here not the natural log. So that's important. Okay see if you can condense number 37 and 38 and then for these last three 39 40 and 41 not the last for the video but just on this page here we're going to use the change of base formula and I'll explain that when we get to those. So what would you do for 37? See if you can try this one. First thing I would do is I'd bring up this 2 as a power. So I'll make this x squared. So let's cross that out. And when we add we multiply. When we subtract we divide. So this would be log base 4 of x squared c. Subtraction we divide. That's d. Now the one-third we can bring that up as a power. Okay so this is the one-third power now. The one-third power is really the cube root. So this is log base 4 of the cube root of x squared times c. all over D and that's fully condensed. Now you can check your work by expanding. You'll get back the original. For number 38 now, this one we've got natural log of A plus natural log of B minus ln C minus ln D minus ln E. How would you condense that one? Well again I would use that little shortcut trick. Whatever logs are positive or added, those arguments are going to be multiplied together. Any of the logs that are subtracted or negative, those arguments are going to go in the denominator. So in this case, a real short fast way of doing this one is natural log of AB over CDE. Now let's talk about the change of base. Now sometimes when you're evaluating logarithms, you might have a base that's not base 10, which is our common log, which just it appears as log on your calculator, or natural log ln, which is your log base e. So most calculators have log base 10 and log base e but what happens if you want to evaluate log base 2 of 11? That's what this change of base formula here is showing us. It's saying that you can pick a new base like for example log base a and it's easy to remember because it'll be c goes in the numerator see how c is a little bit higher it goes up top and b is a little bit lower it goes down below. That's just an easy way to remember it. So for example I could say this is equal to log base 10 which pretty much every calculator has log base 10 on it. Put the 11 up here and I put the 2 down here. And that's it. It's log of 11 divided by log of 2. You want to make sure you don't do log of 11 halves. That's not correct. You have to do log of 11 divided by log of 2. You could also do the natural log like natural log of 11 over natural log of 2. You would get the exact same thing if you do that on your calculator. Number 40 and number 41. Try these. How would you write these as a quotient? Quotient means divided. A quotient of logs or a quotient of natural logs. Basically using the change of base. Same idea. Log base 10. Log base 10. The 12 goes in the numerator. See it's a little bit higher. That's easy to remember. Five's the base. It's a little bit lower. And you would just do this under calculator. Log of 12 divided by log of 5, which is also the same as natural log of 12 over natural log of 5. And one more example. What would you do for this one? Log base 10. log base 10. 8 goes in the numerator, 17 goes in the denominator or you can do the natural log. Natural log of 8 divided by natural log of 17. Remember not the same as natural log of 8 17ths. You have to do them as two separate logs and divide. Okay for these last seven examples this is where everything kind of like culminates and you know basically builds on itself so that we can solve equations using logarithms. So everything we've been using we're going to put into use in these last problems. So see if you can try some of these. I'll show you a few examples. For 42, you can see we've got that variable in the exponent position. That's what we're trying to solve for. I like to think about working from the outside in towards that variable. So what I'm going to do is I'm going to add one to both sides. So that's going to make this 15. Divide both sides by 3. So that's going to give us 2 to the 4x power equals 5. Now here's where, because our variable is in the exponent position, we're going to rewrite it in the logarithmic form. So what I'm going to do is I'm going to take the log base 2 of both sides. These are inverses and they undo each other or cancel. And now we have 4x equals log base 2 of 5. If I divide both sides by 4. Now you don't want to divide just the 5, you want to divide the whole thing by 4. This is an exact answer. Now if you want to get an approximation, we can go to the calculator. Now some calculators you can change the base to base 2. This calculator you can. Otherwise you can use your change of base formula. So this would be, let's see, log base 2 of 5 all divided by 4 is coming out to approximately 0.58 approximately. Okay. So let's try number 43. Now this one, same basic idea. We're trying to get that variable x by itself. I'm going to subtract 2 from both sides. That gives us 10. Working from the outside in towards the variable, I'm going to divide both sides by 5. That's 2. Now because our variable is in the exponent position, I'm going to take the log base 3 of both sides. These are inverses. Now I've got that variable. down from the exponent position so I have a little bit easier getting it by itself. Instead of multiplying by 2, I'm going to divide both sides by 2. This is an exact answer if we want to get a calculator approximation. Let's go ahead and do that on the calculator. So log base 3 of 2 divided by 2. It's approximately 0.32. Okay, approximately. Now for number 44. Let's look at this one now. This one we have a logarithm in here. We want to get that variable by itself. Again working from the outside in I would subtract 4 from both sides. That would give us 6. Divide both sides by 3. That's going to give us log base 2 of x is equal to 2. Now because that variable is inside that logarithm I'm going to rewrite it by exponentiating both sides. Okay and now we have 2 squared which is equal to 4. And you got it. Number 45, here we have a natural log or log base e. So again working from the outside in, I'd add one to both sides, that would be 8. So 2 natural log of x equals 8. Divide both sides by 2, so natural log of x equals 4. Remember natural log is like base e, so I'm going to exponentiate both sides to get that variable by itself, to get rid of that natural log. So we have an exact answer x equals e to the fourth or we can see what that is on the calculator. It's about 54.60 approximately. Let's take a look at a few more examples. Okay last three examples solving equations involving exponential functions, logarithmic functions. Let's see if we can talk about these. These ones we're using our condensing formulas over here. So here what I can see is I'm adding I can actually combine these logs into one logarithm with that log base 2. When you add you multiply. So this is going to be x squared minus 2x. I'm distributing the x equals 3. Now because our variable is inside of this logarithm, I'm going to rewrite it into exponential form by exponentiating or raising both sides using base 2. 2 to the third power is 2 times 2 times 2 which is 8. I'm going to subtract 8 from both sides. and set this equation equal to zero. Now we can factor and set the factors equal to zero. What multiplies to negative eight but adds to negative two? That's negative four and positive two. And then if I set both of these factors equal to zero, you know this from algebra one, you're getting x equals four or x equals negative two. Now remember how we said you can't take the log of zero or a negative number? So if I take this negative two and I put it back in, see how that's negative? So this is actually like an extraneous or false answer. With 4, this comes out to positive 2. This is positive 4. This one's a good answer. We're not taking a log of 0 or a negative. So you always want to just quickly check, you know, just by putting it back in. For 47, how would you do this one? Notice we're subtracting, so you're going to want to condense these into one log. So if I was going to do this one, I would say log base 3 of x divided by 2. See, when you subtract, you divide. And then now because this variable is inside of this logarithm, I'm going to do the inverse which is to exponentiate both sides using base 3. 3 cubed is 27. And the opposite of dividing by 2 is to multiply both sides by 2. So x equals 54. If I put it back in, yep, I'm not taking the log of 0 or negative, so this is a good answer. And then our last problem, we've got our variable in the exponent position. How would you do this one? If I was going to do it, I would just work from the outside in. Instead of subtracting one, I'd add one to both sides. And then here I'm going to rewrite this in the logarithmic form by taking the log base e, which is like the natural log of both sides. Okay, these are inverses and you get 2x equals natural log of 11. And then if I divide both sides by 2, that's our exact answer. Now you don't want to make this 11 halves, it's a natural log of 11 divided by 2. So let's see what that comes out to, natural log of 11. Enter, divided by 2, I'm getting approximately 1.20. And you can check your answer, you can put it back in here and see if it comes out to approximately 10. It's going to be a little bit off because we rounded, but that's it, you got it. So great job if you're able to follow this comprehensive video talking about logarithms. If you want more practice, more examples, you want to see another video I did similar to this video, but different problems obviously. Go ahead and follow me over to that video right there and we'll get some more practice. I'll see you over in that video.

We're going to talk about expanding and condensing logarithms. We're going to talk about the change of base form though. We're going to talk about how to solve equations involving logs. So this is going to be like a comprehensive video going into different aspects of working with logarithms. Let's dive in.

I'm going to have a lot of different examples. You can practice some of these on your own to get some practice. practice. So we'll do multiple types of each kind.

So let's dive into this first set of problems. We're going to talk about how to rewrite logarithms into the exponential form. So the first thing you want to know is this form here. This is the log base B of X equals N. And when we rewrite it, see this is log base B.

This is an exponential function with base B. And notice that here is the power and X is like your answer. So what some students do when they're rewriting is they say, well, I know this is my log base b. I know this is my exponential function base b. All I have to do is interchange these two quantities.

That's one way to do it. I'll show you another way that I like that's a little bit more intuitive. So let's take a look at a couple examples. So for number one, log base 4 of 64 equals 3. Because logarithms and exponential functions, they're inverses of each other, they're kind of like squaring and square rooting or like multiple. multiplying and dividing, they're like, they undo each other, they're inverses.

What I can do is I can exponentiate, I can raise both sides using this base 4. So see how this is an exponential function with base 4 and a logarithmic function of base 4? Those undo one another, they're inverses, and now you can see it's an exponential form. Now I could rewrite this a little bit if I want, I could say 4 to the third power equals 64, that's 4 times 4 times 4, three times, and you've got it. So this is kind of a...

of a little bit more of an intuitive way, kind of like when you had a problem like this, like x squared equals 16, and you say, oh, I'm going to just take the square root of both sides, see the square and the square root, those like are inverses, they undo each other. Of course, the first time you saw that symbol, you were probably like, you know, what is that, right? It's like, and it just means, you know, the inverse or the opposite, right?

So let's go to number two now, log base 5 of 1 is 0. So if I was going to rewrite this, again, you could use this technique that I mentioned here, where you say, oh I recognize this is base 5, all I have to do is interchange these two numbers here, 0 equals 1. Or you can do the method that I kind of like which is to exponentiate. I can raise both sides using that base 5. These undo each other and you can see 5 to the 0 power equals 1. Same thing. Let's go to number three. See if you can do this one.

Now this one's a little bit different. Notice how it says log and there's no base. So what does that mean when there's no base?

Well this is called a common log. and this is understood to be base 10. Now you can write that base 10 in there just to kind of remind yourself and then again you can use the technique that I mentioned here where you keep the base the same base 10 you interchange these two numbers Or you can do the technique that I prefer, which is to exponentiate both sides using base 10. So there's an exponential function. There's a logarithmic function, kind of like squaring and square rooting.

They're opposites. And 10 squared is 100. The negative, you take the reciprocal. That's how we're getting 1. So now it's in the exponential form.

Now you're probably saying Mario, you know, why are we even doing this? You know, why are we switching forms? Well, then we're gonna go through switching from logs to exponential form then exponential to logarithmic form But then we're going to talk about how to evaluate logs, which means how do you find the value, and that's going to come in handy when we get to that section of this video. So see if you can do number four and five on your own. Go ahead and pause the video just to test yourself, see if you're grasping this.

Now again if I was going to do this I would just exponentiate both sides using this base 125. The exponential function and the logarithmic function are inverses and you can see I have 125 to the 1 3rd power equals 5. which makes sense because the 1 3rd power is like the cube root and we know the cube root of 125 is equal to 5. And for the last example what did you get for this one? Well again you can use this technique I was mentioning here where you can keep the base the same. and just interchange these two quantities. So 2 to the 5th power equals 32, which makes sense because 2 times 2 times 2 times 2, 5 times is 32. Or you can use my preferred technique, just because it's a little bit more intuitive, you can exponentiate both sides using base 2. These are inverses, and this makes sense. 2 to the 5th is 32. So let's now talk about going the other direction from exponential function back to the logarithmic form.

Okay for this next group of problems we're going to switch from the exponential form back to the logarithmic form. So I'll show you my preferred method here first. So what I would do is I'd say this is an exponential function with base 3. So I'm going to do the inverse which is to take the log base 3 of the left side, log base 3 of the right side to keep it balanced right, and then the log base 3 and the exponential function base 3, as long as the bases are the same these are inverses they cancel one another out. So now we have log base 3 of 81 equals 4. Or if you want you can flip this whole equation and write it like this. Log base 3 of 81 equals 4. Now that's in logarithmic form.

Now if you want to check your work you can always rewrite it back into the exponential form. You can say oh this is log base 3. How about if I exponentiate both sides using base 3? These would cancel one another out and 3 to the fourth equals 81 which was our original problem. And you might want to do this actually keep going back and forth say okay now it's an exponential form let me go back to logarithmic form and just keep switching just so you feel comfortable changing forms okay. They're equal to either equivalent it's just written in a different way right.

So for number seven let's do this one together so we've got seven to the negative second power equals one over forty nine we know that's true because seven squared is 49 the negative you take the reciprocal okay but all we're doing is we're going to rewrite it in the log form by taking the log base 7 of both sides of the equation. Keep it balanced. And the reason this works is see log base 7, exponential function base 7, those undo one another and now you can see that it's in the logarithmic form.

Now you might want to flip it just so that the logs over here on the left. Doesn't really matter. It's a you know an equation but just to see it from a different perspective. That's now in log form.

So if you want you're getting the hang of this try it number 8, 9, and 10 on your own. And go ahead and pause the video and see if we can match what we're getting here. So 2 to the 0 power equals 1. Of course we know that whenever you raise something to the 0 power that's equal to 1. But in log form let's go ahead and take the log base 2. So you want these guys to match.

Whatever you do to the left side you do to that to the right side. You know that from algebra 1 right? And so these undo each other and now you can see log base 2 of 1 equals 0. I might just want to rewrite it a little bit.

just so it reads a little bit nicer. Log base 2 of 1 equals 0. And one thing that you'll notice is that whenever you take the log of 1, no matter what the base is, that's always going to equal 0. So that's kind of a nice thing to remember. So for number 9, what did you get for number 9? Well let's see. 16 to the 1 half power equals 4. Well the 1 half remember is like the square root of 16 which is 4. But all we're doing is saying let's write it in the log form.

Let's take the log base 16 of both sides. And now you can see it's in the logarithmic form. I'm just going to flip the equation log base 16 of 4 equals 1 half.

And now it's in our log form. Now, again, you can do this another way. You could say, let's say, for example, number 10. What did you get for this one?

I could say, well, here's my base, right? So I could say there's log base 3. And remember how we said we switched the 2 and the 9? So this would be log base 3 of 9 equals 2. If you like that technique, you know, go with that.

Some people, that's more comfortable. They say, oh, I know my base is 3, so log base 3. I just switch the exponent and the answer, and now it's in the logarithmic form. If you don't like that method, you can do my method, which is just kind of like to do the inverse or the opposite.

I would take the log base 3 of both sides, okay? And then as long as these bases are the same, the log base 3 and the exponential function base 3, log base 3 of 9 is equal to 2, you're going to get the exact same thing. So now let's go ahead and talk about... Evaluating logs. Okay, the next 10 problems we're going to talk about how to evaluate logs.

Evaluate just means find the value, find out what the logarithm is equal to, right? So let's start with number 11 here. Log base 4 of 16 equals blank.

Now some students they like to put a variable there, say for example like x. And then what we're going to do is we're going to use the skills that we've already mastered, which is to rewrite log equations into exponential form. So what I'm going to do is I'm going to exponentiate or raise both sides using this base 4. This way the exponential function and the logarithmic function, they undo one another or cancel each other out. So now I just have to answer this question. 4 to what power is 16?

Well, you can see that's going to be 2. So this whole thing, the value of this logarithm is equal to 2. And you got it. Let's try number 12. So this one, log base 3 of 1 27th equals. Now some students... like to put a little question mark.

So they'll say okay how can I solve this? Now you can do my method which I've been showing you is just to exponentiate both sides. These are inverses.

Three to what power is 1 27th? Well I know 3 cubed to the third power that's 27 but because it's the reciprocal this is actually going to be a negative exponent. It's going to be negative 3. For number 13 let's just go with making it equal to a variable x for now. And you can use the technique we were talking about here where we say, okay, this is log base 5. I'm just going to switch the x and the 1 so you're just switching the exponent and the answer and now we just have to answer this question.

5 to what power is 1? That's 0. And remember how we talked about whenever you take the log of 1 no matter what the base is it's 0? And this is this is y right? So for number 14 let's look at this one. Log base 9 of 3 equals what?

What's the value of this log? Let's exponentiate both sides using base 9. You always want to make sure these bases match. Those undo each other. 9 to what power is 3? Well 9 to the 1 half power is 3 because 1 half is like the square root.

So if it was the 1 third that'd be the cube root or 1 fourth that's the fourth root. So the square root of 9 is 3 that that matches. And for number 15 now remember log of a thousand there's not a base here.

So what's the base when there's no base? Base 10 right? That's a common logarithm.

It's a Commonly used on your calculator you'll see log, L-O-G, which some people ask me sometimes, Mario what's this 10G? That's a L, that's a L-O-G, right? But now you just say, okay common log, so this is going to be exponentiate both sides. So these are inverses. 10 to what power is a thousand?

Well 10 times 10 times 10, three times is a thousand, so the answer is three. Let's do five more examples and maybe you can practice these ones on your own. Now before you dive into some of these problems on your own, I want to explain a little something about this Ln. So when you see Ln, this is what's called the natural log.

And Ln, you can write it like this if you want, log base e. So natural log, it means it's log base e. But as a shorthand, we just write it as Ln or natural log.

So again, sometimes it helps to put an e here as a base, just to remind us that it's base e. Just like when we deal with log, we don't see the base. That's understood to be a common log or base 10. So let's do this one. Natural log of 1 is equal to blank. We know this is at log base e, so what I'm going to do is I'm going to exponentiate both sides using base e.

Those are inverses. e to what power is 1? Well, e to the 0 is 1, so this is just going to be equal to 0. So if you want, you can try some of these other ones, 17, 18, 19, and 20, if you want to pause the video. For 17, again, this is...

natural log which is understood to be log base e so I'll put the base there. Notice the base is usually a little bit smaller and it's lower. Okay so that's how you know it's the base. And one thing that's interesting about this particular problem is see how this is log base e and an exponential function base e. So if you can get these to match it's just like we've been talking about using inverses this would just come out to four.

Now if you don't like that method because some students they just like to kind of do it the same way each time which is completely fine. You just say well here's log base e, let me exponentiate both sides using base e. Those are inverses.

e to what power is e to the fourth? Well if the bases are the same then the exponents have to be the same and x would equal 4. So you're going to get the same answer. But sometimes if you can catch it early enough and say oh the bases match, one's exponential, one's logarithmic, they undo each other, it's just a little quicker.

So what did you get for 18, 19, and 20? Well for 18 I notice there's not a base here so this is understood to be base 10. So I'm going to say exponentiate both sides. Those are inverses. 10 to what power is the square root of 10?

Well we know the one half power is the square root so 10 to the one half is the square root of 10. For number 19 what did you get on this one? This one I'm going to do it a little bit differently just to show you. 128 can I get that in terms of base 2?

Well yes because I know that 128 is 2 to the 7th power. Let's see 2, 4, 8, 16, 32, 164, 128. And so now I can see I've got log base 2 and exponential function base 2. And so you can see we're just getting 7. Again you don't have to do it that way. You could just exponentiate both sides. 2 to what power is 128?

That'll be 7. You'll get the same answer. And then for number 20, what did you get for this one? Again, we'll just do the method we've been using, which is to do the inverse or to exponentiate. 1 5th to what power is 25?

Well, 1 5th to the negative 2 power, because we know 1 5th squared is 1 25th, but the negative we take the reciprocal and that's how we're getting 25. Now before you dive into some of these problems on your own, I want to explain a little something about this Ln. So when you see Ln, this is what's called the natural log. And Ln, you can write it like this if you want, log base e.

So natural log, it means it's log base e, but as a shorthand we just write it as Ln or natural log. So again, sometimes it helps to put an e here as a base, just to remind us that it's base e. Just like when we deal with log we don't see the base that's understood to be a common log or base 10. So let's do this one. Natural log of 1 is equal to blank. We know this is a log base e so what I'm going to do is I'm going to exponentiate both sides using base e.

Those are inverses. e to what power is 1? Well e to the 0 is 1 so this is just going to be equal to 0. So if you want you can try some of these other ones 17, 18, 19, and 20 if you want to pause the video. For 17, again, this is natural log, which is understood to be log base e, so I'll put the base there.

Notice the base is usually a little bit smaller and it's lower. Okay, so that's how you know it's the base. And one thing that's interesting about this particular problem is, see how this is log base e and an exponential function base e?

So if you can get these to match, it's just like we've been talking about using inverses, this would just come out to 4. Now if you don't like that method, because some students, they just like to kind of do it the same way. each time, which is completely fine. You just say, well here's log base e, let me exponentiate both sides using base e. Those are inverses.

e to what power is e to the fourth? Well if the bases are the same then the exponents have to be the same and x would equal four. So you're going to get the same answer, but sometimes if you can catch it early enough and say, oh the bases match, one's exponential, one's logarithmic, they undo each other, it's just a little quicker.

So what did you get for 18, 19, and 20? Well for 18 I notice there's not a base here so this is understood to be base 10. So I'm going to say exponentiate both sides. Those are inverses.

10 to what power is the square root of 10? Well we know the one half power is the square root so 10 to the one half is the square root of 10. For number 19 what did you get on this one? This one I'm going to do it a little bit differently just to show you.

128 can I get that in terms of base 2? Well yes because I know that 128 is 2 to the 7th power. Let's see 2, 4, 8, 16, 32, 164, 128. And so now I can see I've got log base 2 and exponential function base 2. And so you can see we're just getting 7. Again, you don't have to do it that way. You could just exponentiate both sides. 2 to what power is 128?

That'll be 7. You'll get the same answer. And then for number 20, what did you get for this one? Again, we'll just do the method we've been using which is to do the inverse or to exponentiate.

1 5th to what power is 25? Well, 1 5th to the negative 2 power. Because we know 1 5th squared is 1 25th, but the negative we take the reciprocal and that's how we're getting 25. Okay, for these next four examples we're going to talk about how to graph logarithms. So let's look at this first example, number 21. y equals log. of x.

Now there's not a base here so we know this is a common log or base 10. Let's rewrite it into the exponential form. So the way we do that remember is we exponentiate both sides okay using the same base these are inverses and so now you can see we have 10 to the y equals x. Now we know that y equals 10 to the x okay looks something like this it's an exponential growth graph. Okay, because see the base is greater than one so it's going to be an exponential growth. But what's the difference between this equation and this equation?

All we really did was interchange or switch the the x and the y and that's how you find the inverse. You're interchanging the x and the y. When you interchange the x and the y, graphically what that does is it reflects the graph over the line y equals x, this 45 degree line, and you get a graph that looks something like this. So let's go ahead and make a table. I'll show you how I normally do this.

I make a table. But what I do is instead of plugging in for x, I'm going to plug in for y since that's in the exponent position. So let's plug in some easy numbers.

10 to the negative 1. Remember the negative exponent you take the reciprocal. So that's 1 tenth. 10 to the 0. Anything to the 0 power is 1. And then 10 to the first.

Anything to the first power is itself or 10. If I plot these points now, 1 tenth down 1 is going to put us right about here. 1, 0 is going to be right here on the x-axis and 10, 1 is going to be right 10 up 1 like this. So you can see our graph is going like... Something like this.

And notice it's getting closer and closer to the y-axis. So we have a vertical asymptote. This is the asymptote x equals 0, this vertical line.

And what happens is the domain is going to be greater than 0. Or if you're using interval notation, you can say from 0 to infinity, like that. And then the range is going to be all real numbers. Because even though this is going to the right, It looks like it's almost horizontal, but it's gradually going up and this is going all the way down to negative infinity So we would say the range is all real numbers or negative infinity to positive infinity. So that's a good technique You can basically rewrite it into the exponential form Substitute in for y instead of x make a table and plot your point So that's a good way to do it.

But looks like at number 22. This one's involving a little bit of uh, Transformations now, what do you think this one does to the graph? And what do you think this 3 does to the graph? Well if you've been studying transformations, you know that the one group of the x has like the opposite effect.

So the minus 1 is actually going to shift right 1, see positive 1, and the plus 3 has the same effect, it's going to go up 3. So what we can do if we want is think of this right 1 up 3 as like our new starting point, kind of like our new origin. I could even draw in like a new y-axis and like a new x-axis. We'll think of that as our starting point.

I'm going to ignore the 1 and 3 for a moment and I'm going to rewrite this in exponential form. So y equals log base 2 of x. I'm going to exponentiate both sides using base 2. And so now you can see we have this equation 2 to the y equals x. Let's make a table, plug in some small numbers, okay, and let's see what we get.

So 2 to the negative 1 is 1 half. 2 to the 0 is 1. 2 to the first is 2. 2 to the second is 4. 2 to the third is 8, etc. Notice I'm plugging in for the y. That makes it a little bit easier to evaluate.

So now if we plot these points, but instead of from our original origin, let's use our shifted origin. So I'm going to go from here, right a half, down 1, which is right about here. 1, 0. So from here, right 1, up 0. Then right 2, up 1. Right about here, and then let's say right 4 up 2. So this would be 2, 3, 4, up 2, right about here. Okay, so there's our graph, and you can see the domain now is going to be greater than 1. So you can say x is greater than 1 but not equal to 1, or if you're using the interval notation you can say from 1 to infinity.

And then the range, the y values, you can see it's going down. Gradually up forever and ever so we can say the range is all real numbers or you could say negative infinity to positive infinity If you're doing the interval notation and then the equation of this vertical asymptote right here remember vertical lines are x equals lines This would be x equals 1 Okay, now another thing you can do too is you can't take the log of 0 or a negative quantity So whatever is in parentheses here, whatever you're taking the log of it has to be positive meaning this quantity has to be greater than 0 So you could solve this inequality, add 1 to both sides. Okay, as long as x is greater than 1, that's my domain. Or if you set it equal to 0, like x minus 1 equals 0, and solve, that gives you the equation of your asymptote x equals 1. Now this is just one way to do it using the transformations.

If you don't like that method, what you can do is just take this entire equation and rewrite it. okay, an exponential form. And the way I would do that is I would work from the outside in towards this x.

So I would subtract three from both sides. So this will be y minus three equals log base two of x minus one, okay. Then I would exponentiate both sides using base two, okay.

So now I just have to add one to the other side. So I get two to the y minus three plus one equals x. Now it's an exponential form.

You can plug in values for y. and you're going to get the output your x. You can make a table.

Just make sure if you use the entire equation that you graph from the original origin, the zero zero point. That's one way students sometimes like to do it. If it's a simple transformation like this, I just like to kind of look at my parent function and graph it from the shifted origin. Let's look at two more examples.

Maybe you can practice these on your own to get some more experience. Okay, if you want to try some of these on your own, try 23. This is a good one to practice. Y equals log base one-third. of x plus 2 minus 1. How would you graph that one?

Well if I was going to do it I would make use of the transformations. I'd say well this number grouped with the x is going to have the opposite effect. It's going to shift left 2. This number that's not grouped with the x that's going to be a vertical shift down 1. So if I go left 2 down 1 I can think of this as my new origin.

I'm going to draw in like a new x-axis and a new y-axis. So I'm going to think of this as my starting point. Then I'm going to focus in on the parent function.

The parent function is just like what's left here, log base one-third of x. And I'm going to rewrite this in exponential form by exponentiating or raising both sides using base one-third. So now I can say, one-third to the y power equals x.

So if y is negative one, one-third to the negative one is three, because the negative exponent you take the reciprocal. One-third to the zero is one, because anything to zero power is one. And then 1 third to the first would be itself 1 third.

So if we plot these points now from this new starting point, let's see, we've got right 3, down 1, right 1, up 0, right 1 third, up 1. You can see this graph looks like this. Now that might look a little strange to you. You might be saying, Mario, I thought you said the law graphs look like this, right?

So what happened? Well remember where we were dealing with our parent function y equals log base one-third of x, right? See if we were to graph y equals one-third to the x power notice that this is a base that's less than one between zero and one that's an exponential decay function. Remember exponential decay functions they go like this down to the right they approach that horizontal asymptote of the x-axis so they look like that.

And remember when you find the inverse, what are you doing? You're finding the reflection of that graph over that line y equals x. So if I was to take that graph and reflect it, it's going to look something like this now, where it's like getting closer, it's going to go kind of like that.

Okay, and that's what we have here. It's just that this graph has been shifted left 2 and down 1. So if you want to focus on the asymptote here, you'd say x equals negative 2. Sometimes they ask you for the equation of the asymptote. It'll ask you for the domain so you can say x has to be greater than negative 2 or you could say an interval notation negative 2 to negative infinity.

And then the range is going down and up forever and ever so you could say the range is all real numbers or you could say from negative infinity to positive infinity for the range. Okay so good. Let's do number 24. Now this one you might need a little bit of help on if you haven't seen this before.

y equals natural log of x. Remember the natural log this is like log base e. So I'll just put the either to remind us.

Let's rewrite it in exponential form by Exponentiating both sides. So we're really trying to graph e to the y equals x. So let's plug in some numbers for y like negative 1 0 and 1. e to the negative 1. Now we didn't talk about this at the very beginning, but this is the natural base e.

Right, and e is one of those numbers kind of like pi. You know how pi is like a non-repeating, non-terminating, like 3.14159, like that. E is about approximately 2.7. So you might want to memorize that. About 2.7.

So if we do e to the first, that of course is going to be itself, 2.7. If we do e to the zero, anything to zero power is one. If we do e to the negative one, that's like one over e.

And 2.7 is almost three, okay? So it's about one third. I'll just say approximately.

1 third. So now when we plot these points, let's do that. 1 third, down 1, 1, 0 means we're going to write 1 up 0, and then 2.7, 1, 2, almost a 3, not quite, up 1. So now if we graph this, there's our graph.

So it's approaching this y-axis, which is the asymptote x equals 0. The domain is going to be x is greater than zero or from zero to infinity if you're using interval notation and the range is going to be all real numbers or negative infinity to positive infinity. But again this is helpful to rewrite it in the exponential form. Okay now we're going to talk about expanding and condensing logs.

These are called the properties of logarithms. You probably remember when you learned about the properties of exponents. Let's go through it real quick. When you multiply and you have the same base What do you do to the exponents?

You add them, right? Similarly, when you work with logs, if you have the log of a product, see x times y, you write it as the sum log base b of x plus log base b of y. So just like when you multiply you add, when you multiply you add.

Just want to make sure you keep the bases the same. This is called the product property of logs. When you go from the left side, To the right side we call that expanding. It's getting bigger.

If you go from the right side to the left side we call that condensing. It's getting smaller. You're combining it into just one log. Now let's look at the quotient property. When you divide with exponents and you have the same base, what do you do to the exponents?

You subtract them. With logs, when you have log base b of x divided by y, what do you do to the logs? You subtract them.

You take log of the numerator minus log of the denominator. This is called the quotient property of logs. And then here, when you have a power to a power with exponents, like an exponent raised to an exponent, what do you do to the exponents?

You multiply them. With logs, when you have log base b of x to the nth power, what do you do with this power? You can bring it down in front of the log and multiply it by that n.

So this would be n times log base b of x. It's also reversible, so you could take this coefficient and bring it up as a power. That would be condensing. When you bring it down, that's called expanding. So this is called the power property of logs.

So we're going to go through some examples together and you can practice some on your own if you want to pause the video. But I'll see if I can show you how this works. Let's do this first example log f u n.

Log of fun right? Fun with logarithms right? Well see here's the thing. They don't tell us the base. That's understood to be base 10. We can see that f u and n are right next to each other.

That means they're multiplied together just like this one here. So we can write it as a sum. logarithms.

So this is going to be equal to log of f plus log of u plus log of n. That's expanded. Now if you if they gave us this we could condense it back into log of f times u times n. So we're combining it into one log.

But in this case we're going to focus on expanding first. Okay let's go to number 26. This one we have log of wx divided by y. Now you can see this fraction bar is like a division sign.

That's like this one, the quotient property. When we divide we subtract. So we could say log of wx minus log of y.

But notice the w and the x are right next to each other. That means they're multiplied together. That's this product property of logs.

We can write it as a sum. So when you multiply you write it as a sum. So log of w plus log of x minus log of y and now it's fully expanded.

Let's go to number 27. This one log base 4 of L divided by MN. So this one we've got a different base. We have base 4. So I'm going to say log base 4 of L minus log base 4 of MN. See remember when you divide you subtract. But notice that M and N are multiplied together.

So what I'm going to do is I'm going to write this as a... Product, okay, you write as the sum of logs. So this would be log base 4 of m plus log base 4 of n.

Now here's where students go a little bit off the tracks. See this minus sign? This whole thing is being subtracted.

That means this whole thing here is a group. So we're just expanding this, but the whole thing is being subtracted. On the next step I'm going to distribute that negative into the parentheses and this is going to be log base 4 of L minus log base 4 of m. minus log base 4 of n fully expanded. Now when we get to the condensing and I'll talk about that a little bit more then but whatever logs are being subtracted those arguments, okay these are that what are called the arguments not like getting into a fight just like they just call them the arguments, the ones that are being subtracted those logs those quantities go in the denominator.

The ones that are positive or added those arguments are going to go in the numerator. So that's helpful to remember. After doing this a while, you'll be able to jump right from here to here.

You'll say, oh, L, that's a positive log. Oh, these ones are going to be negative or they're going to be subtracted because they're in the denominator. So you can kind of skip steps after a while.

But in the beginning, I would just take it step by step. Now for number 28, this one, notice we have an exponent here. This is our power property of logs. We're going to bring that power down in front of the log and multiply it.

So we're going to say 3 times the natural log of x. That's fully expanded. Now if we were going to condense we would bring the 3 back up as a power.

We get back to the original and that's considered condensing, like it's more condensed. Let's look at four more of these expanding examples and then we'll get into the condensing. Okay for these next four examples if you want to try them on your own go ahead and pause the video.

See if you can do them. See how far you can get. Let's dive in number 29. How would we expand this? Well we can see this is log base 5. We can see that the w squared and the z to the fifth are next to each other. That means they're multiplied together.

So we're going to use our product property. So if I was going to do this, I would say this is log base 5 of w squared plus log base 5 of z to the fifth. Now what do we do when we have an exponent? That's our power property.

We can bring that power, that exponent, down in front. and multiply it by the log. So this is just going to end up being 2 log base 5 of w plus 5 log base 5 of z.

And that's fully expanded. Now if you want to check your work you can reverse it, you can condense it. I could bring the 2 back up as a power and the 5 back up as a power and if I'm adding I multiply those arguments together and I get back the original.

So you can check your work on these. For number 30 how would you do this one? Log base 3 of the fourth root of x times y.

Now you can do this a couple different ways. One way to do this is the fourth root is really like the 1 fourth power. So this is like xy to the 1 fourth. Now you can bring the 1 fourth down in front, right?

So let's do that. That's the power property. 1 fourth of log base 3 of x times y.

When they're multiplying, remember that's this product property, you can write it as a sum. So this is going to be log base 3 of x plus log base 3 of y. But that whole thing is multiplied by 1 fourth. Sometimes people just multiply the first one. They forget about the second one.

So you're going to actually want to distribute that 1 fourth into the parentheses. So it's going to be 1 fourth log base 3 of x plus 1 fourth log base 3 of y. That's fully expanded. And again, you can check your work by reversing. those steps and condensing it.

For number 31 we have log base 2 of cat divided by dog. Or you could say the product of cat divided by the product of dog. So here what I would do, this one I'm going to be a little bit shortcutty about this.

Whatever is in the numerator, remember we said that those are going to be positive logs and everything that's in the denominator those are going to be subtracted or negative logs. So this is going to be log base 2 of c plus log base 2 of a plus log base 2 of t minus log base 2 of d minus log base 2 of o which kind of looks like a zero you can't take the log of zero but just know that that's a an o minus log base 2 of g so see how i did that so now you might be saying mario why does that work well one way to think about it is when you subtract subtraction is like adding the opposite right And this minus sign is kind of like a negative 1, right? And this negative 1 you can bring up as a power, that's our power property, right?

So this could be like g to the negative 1. And the negative 1 tells us to take the reciprocal, so that would be like 1 over g. And when we add, we multiply, but if you multiply by 1 over g, that's going to bring the variable down to the denominator. So different ways to look at this when you're doing these problems. Let's do 32. This one... We've got a lot going on.

Okay, so what I'm going to do here is I'm going to rewrite this a little bit. I'm going to say log base 7 of x to the 2 thirds. See how the numerator is the power and the denominator is the root? Times y to the 1 third, okay, all over z to the 4th. Okay, so now that's a little bit rewritten.

So now remember when we multiply, we add, and when we divide, we subtract. So this is going to be log base 7 of x to the 2 thirds. plus, because when you multiply you add, log base 7 of y to the 1 third, minus when you divide log base 7 of z to the 4th.

Now what we're going to do is we're going to use our power property. We're going to bring all these exponents down in front, and we have it fully expanded. So this is going to be 2 thirds log base 7 of x plus 1 third log base 7 of y minus 4. log base 7 of z and now you've got it fully expanded. Okay see if you can do these condensing ones on your own and we'll do four of these easier ones. We'll have a couple harder ones for the last two.

So what would you do for 33? Here you're adding. How could you condense this into a single logarithm?

Well when you add what we're going to do is we're going to go from the right side to the left side. We're going to write it as a product. So this is actually going to be 5 times 3. Or we could just say log of 15. And for the next one, number 34, we're subtracting.

What do you do when you subtract? You divide. So this is really going to be log of 3 over y. Now some students make a mistake.

They'll say log of 3 divided by log of y. You just want to condense it into one log and like that. So you want to make sure it's just one logarithm. For 35, this one's a little bit more challenging. What I would do is I would bring the...

coefficients up as powers first, that's our power property, we're just bringing that up as an exponent, and then when you subtract, you divide. So this is going to be log of x squared minus log of y cubed but when you subtract you're going to divide the argument so this is going to be log of x squared over y cubed. Again just one logarithm.

For number 36 same thing here we're going to bring up the coefficients as powers. Remember the one half power is the square root so I'm going to really write this as a natural log of the square root of w. minus natural log of y squared and then this is a natural log of x to the fourth. Remember when you subtract you divide, when you add you multiply.

The way I like to think of it is all the positive logs, the arguments go in the numerator, all the ones you're subtracting are negative, those arguments go in the denominator. So this would be natural log of x to the fourth times the square root of w divided by y squared. Now another mistake that students sometimes make is they'll take this fraction bar and extend it so that it's the nat even the natural log being divided. You don't want to divide the natural log just the arguments over here not the natural log. So that's important.

Okay see if you can condense number 37 and 38 and then for these last three 39 40 and 41 not the last for the video but just on this page here we're going to use the change of base formula and I'll explain that when we get to those. So what would you do for 37? See if you can try this one.

First thing I would do is I'd bring up this 2 as a power. So I'll make this x squared. So let's cross that out. And when we add we multiply.

When we subtract we divide. So this would be log base 4 of x squared c. Subtraction we divide. That's d. Now the one-third we can bring that up as a power.

Okay so this is the one-third power now. The one-third power is really the cube root. So this is log base 4 of the cube root of x squared times c. all over D and that's fully condensed.

Now you can check your work by expanding. You'll get back the original. For number 38 now, this one we've got natural log of A plus natural log of B minus ln C minus ln D minus ln E.

How would you condense that one? Well again I would use that little shortcut trick. Whatever logs are positive or added, those arguments are going to be multiplied together. Any of the logs that are subtracted or negative, those arguments are going to go in the denominator. So in this case, a real short fast way of doing this one is natural log of AB over CDE.

Now let's talk about the change of base. Now sometimes when you're evaluating logarithms, you might have a base that's not base 10, which is our common log, which just it appears as log on your calculator, or natural log ln, which is your log base e. So most calculators have log base 10 and log base e but what happens if you want to evaluate log base 2 of 11? That's what this change of base formula here is showing us.

It's saying that you can pick a new base like for example log base a and it's easy to remember because it'll be c goes in the numerator see how c is a little bit higher it goes up top and b is a little bit lower it goes down below. That's just an easy way to remember it. So for example I could say this is equal to log base 10 which pretty much every calculator has log base 10 on it. Put the 11 up here and I put the 2 down here.

And that's it. It's log of 11 divided by log of 2. You want to make sure you don't do log of 11 halves. That's not correct. You have to do log of 11 divided by log of 2. You could also do the natural log like natural log of 11 over natural log of 2. You would get the exact same thing if you do that on your calculator. Number 40 and number 41. Try these.

How would you write these as a quotient? Quotient means divided. A quotient of logs or a quotient of natural logs. Basically using the change of base.

Same idea. Log base 10. Log base 10. The 12 goes in the numerator. See it's a little bit higher.

That's easy to remember. Five's the base. It's a little bit lower. And you would just do this under calculator.

Log of 12 divided by log of 5, which is also the same as natural log of 12 over natural log of 5. And one more example. What would you do for this one? Log base 10. log base 10. 8 goes in the numerator, 17 goes in the denominator or you can do the natural log.

Natural log of 8 divided by natural log of 17. Remember not the same as natural log of 8 17ths. You have to do them as two separate logs and divide. Okay for these last seven examples this is where everything kind of like culminates and you know basically builds on itself so that we can solve equations using logarithms.

So everything we've been using we're going to put into use in these last problems. So see if you can try some of these. I'll show you a few examples.

For 42, you can see we've got that variable in the exponent position. That's what we're trying to solve for. I like to think about working from the outside in towards that variable. So what I'm going to do is I'm going to add one to both sides. So that's going to make this 15. Divide both sides by 3. So that's going to give us 2 to the 4x power equals 5. Now here's where, because our variable is in the exponent position, we're going to rewrite it in the logarithmic form.

So what I'm going to do is I'm going to take the log base 2 of both sides. These are inverses and they undo each other or cancel. And now we have 4x equals log base 2 of 5. If I divide both sides by 4. Now you don't want to divide just the 5, you want to divide the whole thing by 4. This is an exact answer. Now if you want to get an approximation, we can go to the calculator.

Now some calculators you can change the base to base 2. This calculator you can. Otherwise you can use your change of base formula. So this would be, let's see, log base 2 of 5 all divided by 4 is coming out to approximately 0.58 approximately. Okay.

So let's try number 43. Now this one, same basic idea. We're trying to get that variable x by itself. I'm going to subtract 2 from both sides.

That gives us 10. Working from the outside in towards the variable, I'm going to divide both sides by 5. That's 2. Now because our variable is in the exponent position, I'm going to take the log base 3 of both sides. These are inverses. Now I've got that variable.

down from the exponent position so I have a little bit easier getting it by itself. Instead of multiplying by 2, I'm going to divide both sides by 2. This is an exact answer if we want to get a calculator approximation. Let's go ahead and do that on the calculator.

So log base 3 of 2 divided by 2. It's approximately 0.32. Okay, approximately. Now for number 44. Let's look at this one now. This one we have a logarithm in here. We want to get that variable by itself.

Again working from the outside in I would subtract 4 from both sides. That would give us 6. Divide both sides by 3. That's going to give us log base 2 of x is equal to 2. Now because that variable is inside that logarithm I'm going to rewrite it by exponentiating both sides. Okay and now we have 2 squared which is equal to 4. And you got it. Number 45, here we have a natural log or log base e.

So again working from the outside in, I'd add one to both sides, that would be 8. So 2 natural log of x equals 8. Divide both sides by 2, so natural log of x equals 4. Remember natural log is like base e, so I'm going to exponentiate both sides to get that variable by itself, to get rid of that natural log. So we have an exact answer x equals e to the fourth or we can see what that is on the calculator. It's about 54.60 approximately. Let's take a look at a few more examples. Okay last three examples solving equations involving exponential functions, logarithmic functions.

Let's see if we can talk about these. These ones we're using our condensing formulas over here. So here what I can see is I'm adding I can actually combine these logs into one logarithm with that log base 2. When you add you multiply. So this is going to be x squared minus 2x.

I'm distributing the x equals 3. Now because our variable is inside of this logarithm, I'm going to rewrite it into exponential form by exponentiating or raising both sides using base 2. 2 to the third power is 2 times 2 times 2 which is 8. I'm going to subtract 8 from both sides. and set this equation equal to zero. Now we can factor and set the factors equal to zero.

What multiplies to negative eight but adds to negative two? That's negative four and positive two. And then if I set both of these factors equal to zero, you know this from algebra one, you're getting x equals four or x equals negative two. Now remember how we said you can't take the log of zero or a negative number?

So if I take this negative two and I put it back in, see how that's negative? So this is actually like an extraneous or false answer. With 4, this comes out to positive 2. This is positive 4. This one's a good answer. We're not taking a log of 0 or a negative. So you always want to just quickly check, you know, just by putting it back in.

For 47, how would you do this one? Notice we're subtracting, so you're going to want to condense these into one log. So if I was going to do this one, I would say log base 3 of x divided by 2. See, when you subtract, you divide. And then now because this variable is inside of this logarithm, I'm going to do the inverse which is to exponentiate both sides using base 3. 3 cubed is 27. And the opposite of dividing by 2 is to multiply both sides by 2. So x equals 54. If I put it back in, yep, I'm not taking the log of 0 or negative, so this is a good answer.

And then our last problem, we've got our variable in the exponent position. How would you do this one? If I was going to do it, I would just work from the outside in.

Instead of subtracting one, I'd add one to both sides. And then here I'm going to rewrite this in the logarithmic form by taking the log base e, which is like the natural log of both sides. Okay, these are inverses and you get 2x equals natural log of 11. And then if I divide both sides by 2, that's our exact answer.

Now you don't want to make this 11 halves, it's a natural log of 11 divided by 2. So let's see what that comes out to, natural log of 11. Enter, divided by 2, I'm getting approximately 1.20. And you can check your answer, you can put it back in here and see if it comes out to approximately 10. It's going to be a little bit off because we rounded, but that's it, you got it. So great job if you're able to follow this comprehensive video talking about logarithms.

If you want more practice, more examples, you want to see another video I did similar to this video, but different problems obviously. Go ahead and follow me over to that video right there and we'll get some more practice. I'll see you over in that video.

Transcript for:Comprehensive Guide to Logarithms

Transcript for:
Comprehensive Guide to Logarithms