you know what exactly is a log and why do we even need logs right well logarithms their inverses of exponential functions it's kind of like if somebody said to you you know why do we need square roots well say for example you want to figure out you know how do I solve this equation x squared equals 100 well you say well the inverse or the opposite of squaring is to square root and then we got x equals 10 right or what's the inverse of multiplying and dividing it what's the inverse of adding subtracting so basically the reason we need logs is because we need a way to do the inverse or undo what exponential functions do and you'll see how that plays out as we go through this video so the first thing you want to know is how do you switch from the Log form which is kind of unfamiliar to us right now to the more familiar exponential form and I'm going to show you a couple different ways I'm going to show you my favorite way first now because logs and exponential functions are inverses what we can do is we can use that to our advantage so if we've got log base B of x equals n what I can do is I can exponentiate both sides ok have this equation using the same base as long as these bases match so that's those are gonna cancel each other out and now we have ax equals B to the N right so now it's in our exponential form we're gonna go through some examples so you can see how this works now if it was in this form x equals B to the N see this is the base this is the exponent what I'm going to do is I'm going to take the log base B of both sides okay to keep it balanced remember in algebra you want to do the same thing to the left the right sides but the log base B and the exponential function with base B those cancel one another out and now you can see we're back to where we started originally so understanding how to switch between the two forms is really key now a lot of teachers okay will teach their students this and they'll say you know this is the base okay this number here it's like a subscript it's a little bit below the line this B and this is the power okay so n is the power that's your exponent and then the X that's your answer or often times referred to as the argument okay not like getting into a fight okay just the argument that's the answer so if you know that's the base that's the power and that's the answer then you can say okay there's my base there's my power there's my answer okay I'm going to show you this method because I think it's a little bit more intuitive and easy to understand so let's go through a bunch of examples here so let's switch this from the Log form to the exponential form we'll do that first so you've got log base 3 of 81 equals 4 so what I'm gonna do is I'm going to exponentiate both sides I'm gonna raise both sides using the base 3 why because exponential functions and logarithmic functions they're inverses just like we talked about earlier squaring and square rooting multiplying dividing adding subtracting when you do that they cancel one another out okay so now you've got three to the fourth power equals 81 which we know is true 3 times 3 times 3 times 3 is 81 okay how about this one log base 2 of 1/16 well again we're going to exponentiate using the same base log base to exponentiate both sides using base 2 those are inverses they cancel so now you can see we've got 2 to the negative 4 equals 1/16 and you've got it let's go down to this one log base 10 so now what I'm going to do is I'm going to exponentiate both sides using base 10 those undo one another and we've got 10 to the one-half power equals the square root of 10 and remember your fractional exponents the denominator is the root so that's why that's the square root of 10 this is 10 to the first power now we're gonna go the other direction okay we're going to switch from the exponential form to the logarithmic form this is the base log okay so this is base 5 so we're gonna take the log base 5 of both sides these are inverses they cancel so we get 3 equals log base 5 of 125 now you can write it either way but I'm just gonna leave that like that for right now this one we've got 7 squared 7 is the base so let's go ahead and do the inverse of exponentiating with base 7 let's take the log base 7 of both sides to keep it balanced these are inverses they undo one another so you can see log base 7 of 49 equals 2 and you got it last one here for this group of problems we're going to take the log base 8 of both sides those are inverses and now you've got 1/3 equals log base 8 of - now if you want to get more comfortable with this process just keep switching back and forth between the two forms what I mean by that is see how you have log base 8 let's just exponentiate both sides using base 8 those are inverses now we get back what we had at the very beginning let's say we're gonna want to take the log base 8 again on both sides right those are inverses and we're back to the logarithmic form and you can keep doing that forever and ever you until you feel really comfortable with switching the forms so that's my preferred method just because I feel like it's more intuitive but if you don't like that method stick with what you like some teachers will even just use the circle technique they'll say this base raised to this power equals that answer but just remember that's the base that's the power of the exponent and then the to the argument is the answer so the next step in logarithms we're going to get into let me erase this board and we'll get right back at you okay now we're going to talk about evaluating logs so when they say evaluate what does that mean it just means find the value in the last part that we were talking about we were switching forms between the exponential and logarithmic forms but now what we're doing is we're actually solving we're actually getting an answer and what you can do is you can do the same thing like we were doing in the last part just go ahead and exponentiate both sides of the equation so the exponential function logarithmic function they cancel one another out as long as the bases are the same so now what we have is 2 to what power equals 32 well you can see that's going to be 5 now oftentimes what I recommend to students is just put a variable here like X for example so log base 5 of 125th equals x right so we're not as familiar of thinking in terms of logs so let's go ahead and exponentiate both sides make sure that these bases are the same their inverses they cancel each other out 5 to what power is 125th negative 2 because 5 squared is 25 but the negative exponent takes the reciprocal right this one here log base 49 of 7 again we're just going to make it equal to a variable X we're going to exponentiate both sides right so those are inverses 49 to what power is 7 1/2 because remember the 1/2 power that's like taking the square root so the square root of 49 seven and then I just wanted to make a note here before I get to this next one is that when you see log base 10 of X a lot of times they're just gonna write it as log X they won't even write a base here it's called a common logarithm it's kind of a lazy man's way of just you know writing log base 10 of X you'll see that on your calculator log X but that's base 10 and then log base E of X remember E is the natural base e 2.71 right they'll write that as Ln X or natural log of X so these are like shorthand so if you see that if you want you can put an e there to remind yourself or a base 10 there again to remind yourself what base it is so for this one log base 3 of 1 again we're just going to make it equal to ax let's exponentiate both sides these are inverses 3 to what power is 1 well anything to the 0 power is 1 so that's just going to be 0 so now let's get into the graphing so how do you graph a log okay well here's the thing we know that exponential functions which look like something like this this is an exponential growth function and logarithmic functions are inverses but when you want to find the inverse of a graph what you do is you reflect it over the line y equals ax so if we do that the logarithmic graph should look something like this right here now the exponential graph notice it's getting closer and closer to the x axis whereas the logarithmic graph is getting closer and closer to the y axis so this is going to be our vertical asymptote as opposed to when we had the horizontal asymptote let's go through some examples so you can see what I mean so when you have y equals log base 4 of X you want to graph that what I would do what I recommend is just exponentiating both sides okay so now what we've done is we've rewritten it in exponential form so we've got x equals 4 to the Y power pick some values for Y now as opposed to X because it's a little bit easier to take 4 to the negative 1 that's going to be 1/4 4 to the 0 is 1 4 to the first is 4 and 4 to the second power is 16 now what we can do is we can plot those points so we've got let's see when X is 1/4 we're going down 1 when X is 1 we're going right 0 and we're right for we're going up one and if we go way over here 16 we're gonna be up at 2 so I'm kind of going off the graph there but our graphs going to look something like that ok so now the next thing your teacher might ask you is you know what's the equation of the asymptotes well because this is approaching the y-axis that's actually the line x equals 0 so that's your vertical asymptote they might also ask you what's the domain and the range well remember the domain is whatever the X values can be and in this case X is greater than zero you don't see the graph over here in the negative range it's not touching the y-axis so X can't be 0 so the domain is X is greater than 0 and then for the range we think about what the Y values are and you can see this is going down towards negative infinity gradually making its way up towards positive infinity so the range is going to be all real numbers ok let's do a little bit more complicated example so this one here what do you think the one in the three do to the graph well the number that's grouped with the X and it shifts the graph in the X direction the horizontal direction but this minus one has the opposite effect it's actually gonna shift it right one so we actually have a vertical asymptote here at x equals one okay and then what do you think the positive three does well it shifts it up three so we can think of this point right here that circle that I drew as our origin as our starting point now we're just going to focus on the parent function okay y equals log base two of X we're going to exponentiate both sides okay these are inverses so we get two to the y equals x so I'm gonna make a table again I'm going to plug in some easy values for y so this is going to give us one half one two and four and then what I'm going to do is I'm gonna plot the points from here not the origin because everything's shifting right one and up three anyways so let's just start from here and let's see so if we go right a half down one that'd be right there right one okay up zero and then right - okay let's see right - up one like that and then right four one two three four up two so like something like that so basically this graph looks something like that all right and you can see the domain is going to be X is greater than one and the range is gonna be all real numbers so it's going up and down from negative infinity to positive infinity okay so that's talking about the graphing the next thing we're gonna talk about is expanding and condensing logs so let me erase this board and we'll do that next okay now we're gonna talk about expanding and condensing logs but first we want to talk about the properties of logs okay just like there's properties of exponents rules of exponents for example when you multiply and you have the same base what do you do to the exponents you add them right when you divide and you have the same base what do you do to the exponents you subtract them what are you doing you have a power to power you multiply so those are the rules of exponents you learned before but now the rules of logs it's very similar like when you have these two arguments multiplied together you can write it as a sum of two logs the main thing is you want to make sure that you keep the base the same whatever that base is same thing over here if you're dividing these two arguments you can write it as a difference of two logs just take the log of the numerator minus the log of the denominator and then this one's called the power property you can take this exponent here and you can bring it down in front and write it as a coefficient now all these properties they're reversible so when you're going from the left to the right that's called expanding it's getting larger but if you go from the right to the left that's called condensing it's getting filled more condensed right so if you see the sum of two logs you can write it as the log of a product or if you see the difference of two logs and they have the same base you can write it as the log of a quotient okay and then there's another format called the change of base formula and this really comes in handy when some calculators they don't have the ability to change the base they only have log base ten or a natural log which is log base E so what you can do here is if you have log base B of C equals log you pick a new base let's say base a you take log base a of C divided by log base a of B so for example if you had log base three of seven you could really write this as a common log like log base 10 of seven divided by log base 10 of 3 now it's easier to remember because the seven is a little bit higher that goes in the numerator the three is a little bit lower that goes into 9 you just pick a new base and that's it you can also write this as natural log of seven over natural log of three same thing with this one log base four or five I would just say natural log of five over natural log of four put that in your calculator and you got the same thing as log base four of five but let's get into the expanding and condensing now so say you want to expand log of a times C times D I didn't write the base there so that's considered a common log or log base ten but because these are all multiplied together we're going to use the product okay rule so this is going to be log of a plus log of C plus log of D pretty easy right here we've got a combination of multiplying and dividing so what I'm going to do is I'm going to do log of a plus log of C minus log of B so the one that I'm dividing by we're subtracting the ones that are multiplied together we're adding this one here natural log this one's going to be I'm going to do it in two steps going to do natural log of X minus C because we're dividing natural log of Y times E but you can see these are multiplied together which we can write this as natural log of y plus natural log of Z right but you see this whole thing is being subtracted so we have to distribute that negative so this whole thing comes out two natural log of X minus natural log of Y minus natural log of Z the thing to take away from this particular example is whatever's in the denominator you're going to be subtracting whatever is in the numerator is going to be positive are going to be adding okay so that's just a quick way to figure these ones out but you can do it in steps like this to this one we've got some exponents here so this is actually going to be natural log of x squared plus natural log of Y cubed because you can see these are multiplied so I wrote it as a sum of two logs then what I'm going to do is going to bring those powers down in front as coefficients that's the power property okay for logs so this is going to be two natural log of X plus three natural log of Y that's fully expanded and then one here this is pretty a more challenging one let's see it's log of let's see log of 5 plus log of plus two log of YC I brought down that two in front okay and then - because we're dividing one half log of W minus one half log of Z so I did all that in one step now you don't have to do it in one step like I did I just want to show you a more challenging one but notice everything that's in the denominator is being subtracted the square root is the one-half power so I brought that one half down in front same thing with here the Y squared I brought that two down in front everything that was in the numerator that was multiplied together I wrote it as a sum of logs and the ones are in the denominator I subtracted okay that's the quotient rule now we're going to go over to the condensing so we want to condense these into one log so we're subtracting so what I'm going to do is I'm going to divide and don't get confused you want to make sure the one that comes first or to the left is in the numerator the one you're subtracting that one goes into the denominator so it's like natural log of 2 divided by X this one with the coefficients I would bring those up as powers first and then you can put it all together so this would be natural log of X cubed times y to the fourth okay adding so we multiplied okay and then this last one we've got quite a bit going on here so I'm going to bring up the coefficients okay one half is going to make that the square root of Z so if I do this is going to be log of X to the fourth right over see because we're subtracting y squared plus that's going to be times the square root of Z so that was a challenging one so this has been expanding and condensing logs now we're going to get into evaluating equations and solving equations using logarithms so let me erase this board and we'll get back with that okay now we're gonna solve equations using logarithms so just like algebra you want to work from the outside in you want to try to isolate that variable same thing here we're trying to get that X by itself so let's go ahead and subtract 7 from both sides so that gives us 4 times 2 to the 3x power equals 3 again working from the outside in let's divide both sides by 4 so now we have 2 to the 3x equals 3/4 whenever that variable is in the exponent position okay you need to get it down from there we need to reverse that exponentiation process we want to take the log base two okay of both sides of the equation to keep it balanced right so the log base two and the exponential base two those cancel so now we have three x equals log base two or 3/4 if we divide both sides by three that's it you've got it now you can do that in your calculator I did that here for us it's a negative 0.138 approximately okay let's go to an example number two same thing here you're trying to work from the outside and you're trying to get that X by itself so let's add 8 to both sides so that gives us 2 log base 2 of x equals 14 let's divide both sides by 2 we're still working from the outside in we've got log base 2 of x equals 7 we want to get that variable by itself what's the inverse of taking the log base 2 exponentiating using base 2 so we have x equals 2 to the seventh power which is a hundred and twenty-eight okay an example number three here we're going to use the properties of logs to condense these into one log remember when you're adding what do you do to the arguments you multiply them so that's going to be x squared minus 2x I'm multiplying those together equals one and then again we're trying to work from the outside in we're trying to get that variable by itself so let's exponentiate both sides so I'm raising base three so those cancel one another out we get x squared minus 2x equals three I'm going to subtract three from both sides set this equation equal to zero I'm going to factor okay so is this all coming back to you from earlier in math right and then now if we set these factors to zero we get x equals three or negative one but the one thing you want to be careful with with log equations because remember the graph looks like this right well you can't take the log of zero you can't take the log of a negative number that's not in the domain okay so if we look at negative one so you'd be taking a log of a negative one a negative number that's extraneous that's not possible three is okay we're not going to be taking the log of zero or a negative value when you simplify that argument okay so that has to be a positive value once you simplify so it looks like we just have one answer x equals three and you got it so we did quite a bit here in this video giving you all you need to know about logs from evaluating to solving equations to expanding condensing graphing I hope you like this video subscribe to the channel check out more math tutoring videos on my youtube channel Mario's math tutoring and I want to mention if you're preparing for the AC T or the SAT check out my huge AC t-- math review video course or my huge SAT math review video course pack with tons of information if you like my teaching style you'll get a lot of those courses and I look forward to seeing in the future videos I'll talk to you soon