so welcome to the 1.07 lesson today we are going over logarithmic functions so by the end of this lesson you will be able to solve logarithmic functions identify the key parts of log graphs and use log functions to model applications and our agenda is on the screen we're going to start by reviewing the properties of logarithmic functions in case you forget those we'll look at transformation of the graphs applications of logarithmic equations and then at the end we'll talk about the natural log so jumping right in here is the formula that is given to you in your online lesson of a logarithmic function it is a function given by f of x equal to log base a of x and it's actually the inverse of the function we talked about last class it's the inverse of exponential functions so you're going to see some similarities because of that and here on the screen we've got some applications of why we learn about logarithms and how we can apply them so in this first picture any ideas what this might be representing type in the chat box good guess ethan it's not a hurricane though this is actually coming from the crest of the earth we don't have a lot of them in florida so that's why this one's really rare yeah brenna it's an earthquake so the richter scale how we measure the magnitude of an earthquake is actually a logarithmic function what about this application any ideas of this pool it looks like they're measuring something in the pool water yeah jordan nice the ph the ph of the pool water ph um the ph scale is also a logarithmic function so how acidic or or basic a solution is any ideas for the third one you're on the right track travis it is loudness how do we measure that ah andrew's got it decibels the decibel scale how we measure how loud something is that is logarithmic and then this last one is yeah jordan password strength so you know sometimes we get annoyed when we have to make a new password and it's telling us we need to use certain characters and it'll show us you know if we have a weak one or a strong one that's all based off a log rhythmic scale too and yeah definitely for a good reason so lots of reasons where we might apply this in the real world so let's get on with it so as i said before the logarithmic function is just the inverse of the exponential function and so we're going to begin by reviewing how do we write logs into exponential form so if we have this logarithmic function here log of 4x equal to 3. first thing we want to review is that if there isn't a base if there's not a base number here then that means it's the common logarithm it just means that there's a base of 10. so we can write the 10 in there if we want but if it's not there we just have to know that that's a 10. and log base 10 is the common logarithm that's what the button on your calculator is it just says log on your calculator but it's representing log base 10. and so to review how we convert logarithmic form into exponential form the easiest way i remember is using the counterclockwise circle method so we take our base we start at our base and we take it to the power of whatever is on the right side of our equation so 10 to the third power then we set it equal to and we continue that counterclockwise circle we set it equal to whatever is in the in within our log function so log base 10 of 4x equal to 3 is the same as 10 cubed equal to 4x and again that would be the same if it was written just as log 4x equal to 3. we know that even though there's not a number there it's 10 so 10 to the third equal to 4x and so this is going to be really important that we can review how to convert because we'll take these logs and we'll convert them into exponential forms so that we can then solve them so if you go back to desmos you have this question where you're going to select all of the following statements that are true and so you'll check each one of these and i've given you a log function and then i've given you an exponential function and you have to check to see are they the same did we convert correctly so i'll give you a moment to go through those and i'll put that link back into the chat box again in case you have not joined the desmos or you accidentally logged out of it so go ahead i'm still missing a bunch of people in the desmos so please join me there good i've got lots of people selecting correct answers i'm putting that link in the chat box one more time because i still have a handful of people who are not in the desmos please join in so you can participate remember you came to the live lesson to get some extra practice this is going to help all right i'm going to give you 10 more seconds and then we will go over the solution so if i have any volunteers that want to share one that they think is true and why please raise your hand all right jordan f you were first which one of these are true and why uh it's the second one log five x equals two because when you think if it's just log and there's no number next to it like if it's log three for example if it's just log it's log base 10 and so you would do the counter clockwise method flip them around and it would be 10 exponent 2 equals 5x that was a wonderful explanation i couldn't have said it better myself thank you so much jordan all right um second we've got jay do you have another one that is true yes um i am gonna choose the first one that one's correct um because uh the log 3 you put that to the power of 4 so it's 3 to the power 4 and then it's set equal to x so it's 3 to the power 4 equals x which is what they have there nice job what about another one do we have another one that is true alejandro and the testing distinction okay i would want to mark the the last option at the bottom because the 10 to power 3 is equal to a thousand which happens to also be the same result as a logarithmic value of a thousand that equals three because uh ten to the power of three over here there's a three over to the zero so there'll be like uh three zeros around ten or just a thousand and because three is the power to raise ten up to a thousand it's considered as a logarithmic value nice job and i like how you explained that one conceptually and this one was maybe a little tricky because i gave you the exponential version first but nice job realizing that it doesn't matter which order we put them in just like it wouldn't matter if let's say this logarithmic was written backwards and we had 3 is equal to the log of 1000 that still means the same thing we would still have the base of 10 down here you just have to be careful if it's written like this then instead of doing it counterclockwise you would be doing a clockwise circle good and then if we look at the third option nice job i see everyone put their hands down because this one is not correct it is not true if we were to check this should be six to the third power equal to 4x so they just messed up the order on that one so important that we review converting those like i said because now we're going to use it to solve so for our second practice go ahead and stick with me here and zoom we're going to solve this one together and they tell us solve log base 5 of 100 minus log base 5 of 4 equal to x so in this problem when they say solve what are we solving for right we're solving for x so we've got this equation we've got x over here so we need to to simplify the left side of our equation and isolate the x so up here in the right hand corner i've pasted the logarithm properties again this is from your online lesson hopefully these should be a review but i've put them up here just so we can use this to help us solve so in looking at our logarithm and we're looking at our left side i would i see that i have two logs that are being subtracted and they have the same base of five so any ideas looking at our review up here which property will i use to simplify this nice hannah's got it since i have two logs being subtracted and they have the same base i can apply this quotient rule here which means instead of writing them as two logs being subtracted i can actually write them as a quotient where my base stays the same and then in the numerator is what's inside the first log the denominator is what's inside the second so i would have log base 5 of 100 over 4 and then i still have equal to x i haven't done anything to the right side of my equation so we can easily simplify that 100 divided by 4 is 25 and so now i have one logarithmic function and here's where i'm going to apply the what we just practiced of now converting it into an exponential form so that i can solve so we're going to use that fluke method 5 to the power of x must equal 25 and now we have an exponential equation and this is what we covered in lesson 1.06 so 5 to the x i know that 25 is the same as saying 5 squared now that those bases are the same i know x must equal 2. not so bad and i do have the typed up version in case you wanted to screenshot that let's try another one and this time you're going to solve this one and if you go to desmos you should be able to actually write on the desmos and show your work so they want you to find x if 7 log base 3 of 9x is equal to 42 and again i have those properties reviewed for you up on the screen and i'm going to give you a minute or two so you can take the time and actually solve this one on your own nice i see olivia has the correct first step thank you to those of you who are showing each step i love that it's good practice because on those essay questions like you have on your mid module checks and your module exams you need to show every step nice answer matthew so in this problem we're trying to solve for x meaning we need to isolate x we need to get rid of everything else so we know that we're going to use the log rhythmic function and we're going to transform it into an exponential but before we can do that we have this 7 being multiplied out front it's being multiplied to our logarithmic function so what can you do to get that 7 out of there that's going to be your first step good your classmates are giving you hints in case you're stuck i'm giving you a little bit more time just because i do see that people are working hard i don't want to cut you short also i love all the work that i see with i actually see you guys putting your circle in there all right so let's look at this one together now and then i'm going to share out some of the work that i like so on this log like i said we're solving for x we're trying to isolate the x and we know that with the logarithm we're going to transform it into exponential form so that we can solve but like i said this logarithm is actually being multiplied by 7. and so before i can make my circle and convert it into an exponential i need to get rid of this 7 and so remember inverse operations if i'm trying to isolate x and 7 is being multiplied to this i'm actually going to divide both sides by 7 so that that will cancel out of the left-hand side so on the left i'll just be left with log base 3 of 9x and 42 divided by 7 i would have 6. so that was the first step and i know a couple people were getting stuck there but now now we just have log base 3 of 9x equal to 6. now i can convert this into an exponential form so 3 to the sixth power must be equal to 9x and here i'm going to simplify this one 3 to the sixth power i plug that in my calculator and i got 729 is equal to 9x and again i'm trying to isolate x so since it's being multiplied by 9 i need to divide 9 on both sides which will leave me with x is equal to 729 divided by 9 and again i used my calculator and i got 81 for that solution so pat yourself on the back if you got 81 pat yourself on the back if you attempted this problem it's okay if you didn't get the right answer or you got stuck at this step we're here to practice so don't be too hard on yourself i did want to share some work that i thought was pretty outstanding so i'm not sure if you can see that hopefully you can clear my annotation um this was work that a student submitted and i really liked this because if this were an essay question this student showed every step they showed that they were divided by seven at first here was the result of that step then they converted into exponential form simplified then divided by 9 to get the final answer so i love that how they showed every step of their work right there and that's like i would like to see on your your essays and just make sure you are using the math type tool that will help you show your work nice and clear and as well as properly and then i did want to share another example this one is good they got the correct answer but it could be even better if um if it were written in step format like the last one so they did the correct work they did divide the 42x7 but when it's written all by itself it just looks like it it's isolated we can't tell that it was part of the equation so that would be my only improvement for that one but otherwise you guys did really well on those and here's the typed out solution so we're moving on now to look at the graphs of logarithmic functions and as i stated earlier the logarithmic function is the inverse of the exponential function so what's really cool here is if you look at this graph f of x the blue line is an exponential and that's that basic exponential we looked at 2 to the power of x and if logarithmic functions are the inverse then we know that they're going to be reflected across the line y equal to x and so here you'll see the orange line is our log function and that is log base 2 of x which is the inverse of 2 to the power of x and so you'll see that when we did exponentials last week we had that horizontal asymptote at y equal to zero right well if we're taking the inverse for logarithms instead of a horizontal asymptote you're going to see a vertical asymptote at x equal to 0. and so that's a big characteristic because now instead of horizontal we've got the vertical asymptote and you'll notice that the domain and range are actually flipped from what we had in four exponentials so the domain of a log ranges from zero to infinity and because of that vertical asymptote 0 is not included it gets really close to 0 but it doesn't doesn't cross 0. whereas the range will be all real numbers it's going to go from negative infinity to positive infinity and if you look back at the domain and range for your exponentials they were actually switched and so we'll talk about inverse functions more next module but they're really cool how they are related and so you can see all the other characteristics of the logarithmic graph i'm put this slide up so you can review the transformations but remember that transformations are the same for all functions so it's really awesome news that the same transformations we've been looking at at all the other functions or types of functions that we've covered so far these are the same but i did put it up here to review so you can see how it's going to affect our logarithm where if there's a negative out front of our function that's reflecting across the x a constant being multiplied is either going to stretch or shrink it if the negative is inside the function that's reflecting across the y if you're adding or subtracting a constant inside the function that's going to be your horizontal shift and remember that's always opposite of how you think it'll act if it's you're subtracting a number that's actually going to move it to the right whereas if you're adding a number it's going to move it to the left and then if you're adding or subtracting a constant outside of the function that's your vertical shift positive will go up negative will go down another great thing about when we're looking at graphs is remember you're allowed to use your graphing calculator so you're always free to check your answers using that graphing calculator so going back to desmos this one is a nice easy multiple choice one i just need you to identify the graph g of x equals log of x minus three plus seven and so if you think about that graph of just the regular log function that i showed a couple slides ago how might that be transformed and yes andrew the transformations can be applied for for any of the logs regardless of what the base is uh good question so b is the base and you're not going to have transformations on that base however the b value that base value could change but b is that base is always going to be a positive integer and remember if it's not written like here then we know it's 10. it looks like you guys have got this you've got transformations down does anyone want to take the mic and tell me what the correct answer is and how you knew jay i saw your handles up first but you've already answered today so if you don't mind i'm going to give them mike to is it carry alice nope did you change your mind no i got it it's kiara elise cara lisa i really apologize thank you for correcting me which one is the correct answer here the correct one is b good why do you say so so first i looked at the um x minus three and since it's opposite i went to the right side three times and then since it's plus seven on the outside i went up seven times wonderful and like i said if you weren't sure and if you look b and c are very similar you'll just notice that c went to the left three if you can't remember or you just want to verify your answers correct graph this function plug this equation into your graphing calculator and make sure that this is the correct one and you'll see that she was exactly correct it is right and the reasoning was beautiful all right so looking at that same equation that same function the same graph my follow-up question is what is the domain of this function so when you figure it out go ahead and type it into the chat box what is the domain and make sure you're putting either parentheses or brackets on the outside so we know if those values are included or not included very nice job and you guys have the correct parentheses as well so domain we're going to keep talking about domain and range all year so domain is asking us about our possible x values and we can use the graph to help us here we can see that we've got because we've translated this function right three values that vertical asymptote is now at x equal to three so this is a vertical asymptote we know it's going to get really close to 3 but it's never going to touch and so that would be our lower bound and it has a parenthesis because 3 is not included and then we see that this graph is going to keep going and going and going up to positive infinity so that would be our upper bound and infinity always has a parenthesis since we can't actually get there very nice job thank you for all your help in the chat box you guys are on it this morning all right let's look at an application problem so for this practice they tell us a chemist uses a ph scale to determine how acidic or basic a solution is when a solution has a ph level below 7 it is considered to be an acidic solution the ph can be calculated using the equation ph is equal to negative log of h plus where h plus is the hydronium ion concentration if the solution has a hydronium ion concentration of 5.6 times 10 to the negative 3 moles of solute per liter or moles per liter is the ph considered acidic all right and naomi knows exactly where i'm going it's a word problem so the first thing i'm going to do is identify what have i been given what am i trying to find and then i'll worry about actually doing the calculation i know some of you are like oh no this is taking me back to chemistry and maybe that's a good thing maybe that's a bad thing but the good news is in pre-calc we love the application problems because it's why we're learning the math and we're going to break it down together to make it easy so for our givens they tell us the ph scale determines how acidic or basic a solution is if the ph is less than seven then that is considered an acid they also tell us the function four ph is negative log of h plus where ph is whether it's acidic or not and h plus is the hydronium [Music] ion concentration if the solution has a concentration of so they're actually giving me the h plus value of 5.6 times 10 to the negative 3 moles per liter is the ph considered acidic so what are we trying to find here because the question is asking us is the ph considered acidic well if i do my math that's not going to tell me acid as an answer they're yeah they're they're wanting us to calculate the ph level and then based on the answer we get for the ph we can then compare is it less than seven or not all right so now that we've gone through that part let's move on to our work we're going to use the formula the function given to us ph is equal to negative log of h plus but instead of using the variable h plus i know what that value is so let's plug in 5.6 times 10 to the negative 3. and so for this one we have what is our base here of this function of this log yeah we know it's 10. this is the common base and what's really great about the common base is we have a button for it on our calculator and so i'm actually just going to use my calculator to calculate this so i'd say ph is equal to and in my calculator i would say negative then i would hit the log button and then i would type in 5.6 times 10 to the negative 3. and if you aren't sure how to do the scientific notation in your calculator that might be something that you work on with your teacher after class you can also always convert this into a decimal and move your decimal place over 3 and you could plug in 0.0056 instead but yeah you'll definitely want to practice because some calculators you hit log first and then what's inside the parentheses some calculators you put what's in the parentheses first and then hit the log so just make sure you know how to use your own calculator but nice job i'm seeing karma and nicole already got those correct answers when you do use your calculator you will get 2.25 as your ph so then we compare that they told us that if the ph is less than 7 it will be acidic so is this solution acidic yes it is acidic because that ph is less than seven yeah it's just quite acidic and yes you can you can convert and actually write it as a decimal if you'd like so you either write 5.6 times 10 to the negative 3 in the parentheses or that is the same as 0.0056 all right here is the nice neat version all right in our last section we are going to talk about the natural logarithmic functions so if you think back to our last lesson when we were looking at exponentials we talked about the the natural exponential e to the power of x so it only makes sense that if we're talking about the inverses of exponentials today we're going to talk about the inverse of the natural exponential as well and the inverse of e to the x is log base e to the x and you can see the graph here in orange but this is actually a special log because log base e is actually the same as what we call the natural logarithm function and again you have this button in your calculator it's an ln and the great thing about um this is again those those transformations remember those apply to all functions so they apply to the natural logarithm function as well so here's the formula that is from your online lesson again just stating that if you have log base e to the x we write that as ln of x it's known as the natural logarithmic function and so this is important because we'll use this to help us solve equations especially when there is that e value involved so for number seven we're going to do this one together in zoom they want us to solve 2 times e to the power of 3x equal to 8. and again when they say solve they just mean find x what is the value of x so up here in the corner of your screen i've copied those properties on the review of of ln properties from your online lesson there is one that i want to add in here though that will help us solve this problem and it is the inverse of this this one so the one i want to add in is that ln of e to the power of x is just equal to x just like e to the power of ln of x the e and the ln cancel out you're just left with x inverse is true here the ln of e will cancel out you'll just be left with x so we're going to use that to help us solve this equation any idea is my first step just type it in the chat box if i'm solving for x i am isolating the equation i want to try and get x all by itself on the left hand side good we are going to divide 2 on both sides that would be my first step that's going to get rid of the 2 and leave us with just that e to the 3x equal to 4. now i'm going to apply this property like you said in the chat box we're going to apply the ln and if you think back to last lesson when we were solving exponentials some of them we had to apply the log to both sides we can apply the natural log to both sides as well so ln of e to the 3x equals ln of 4. and just remember whatever you do to one side of the equation you have to do to the other but this is helpful because that actually gets rid of our e because remember ln of e to the exponent of anything is just going to equal the exponent so this left side is now just 3x equal to the ln of 4. and so then my final answer i would divide by 3 on both sides my final answer would be ln of 4 divided by 3. and i left it here in exact form you can plug this into your calculator you have an ln button you could say ln of 4 divide by 3 and you would get a decimal answer but typically on your multiple choice options they're going to keep it in exact format they're not going to do a decimal because the decimal is rounded so there's your nice typed out answer and yes if you did put it in your calculator to get a decimal it should be about 0.462 if you want to check and just make sure you you can use your ln button all right we've got one more example to go through today and it is going to be using the change of base formula so when we're using logarithms we know that if we have a base of 10 that's really great because we can use our calculator but if the base is any other number then we have to convert it into exponential form and try and solve that way however there is a formula that can help us evaluate using our calculator and that is known as the change of base and so that formula is if you have log base a of x you can actually rewrite this as ln of x divided by ln of a and what's great about this is we can actually plug that into our calculator we can evaluate a log with a non-common base using our calculator and so i give you an example down here of how you would rewrite that whatever is inside the log will be in the numerator whatever the base is would be in the denominator so we're going to use this to help us solve our last practice problem and if you wouldn't mind going back to desmos for the last practice all i want you to do is tell me what would be your first step here they want you to determine the y-intercept for this given function so if you swap back over to desmos what is your first step and it i'll give you a big hint here it doesn't matter what function i put in here no matter how easy or how complex i make it your first step to find y-intercept is always the same so be careful this one we're not solving for it doesn't say solve it says determine y-intercept good i'm so happy to see a lot of you putting the correct answer written in different ways but still correct and yes tyler you're right we're going to set x equal to zero so remember when you're finding y-intercepts you set x equal to zero and you solve for your y value opposite if you're finding the x intercept if you're finding the x intercept you set y equal to zero and you solve for x so our first step would simply be to plug in 0 for x so we've got log base 6 of 0 plus 3 plus 4. now we can simplify what is in the parentheses 0 plus 3 is just 3. but i can't add this 4 to it be really careful because i know it's easy to think that we can simplify this since these are both constant numbers that we could just add them but these parentheses signify this is the logarithmic function so this 3 is actually inside this logarithmic function the 4 is being added to the outside so you cannot add those they need to stay separate and so here is where we're going to apply that change of base we've got log base 6. so i know i can't put that in my calculator but i can if i change the base so instead of log base 6 of 3 i'm going to use get ln of 3 over ln of 6. plus 4. and again we could put that in our calculator we could get a decimal answer but in this case i'm actually going to keep it in ln format so that i have the exact answer and so my y-intercept would be zero comma ln of three over ln of six plus four oops and that would be my answer and one thing i want to point out about the change of base you could actually use log as well log base 10 because it's a proportion here and if you do log base 10 of 3 and log base 10 of 6 you're actually going to get the same proportion as ln of 3 divided by ln of 6. and you can check that on your calculator see if it gets you the same answer it should you should get a decimal of like 4.61 if you type that into your calculator so matthew that's a good question technically that would be correct um i'm showing you how to take it even steps further because if this were a multiple choice question your answers would be in a format similar to this whereas technically you're right i could say that this is my y-intercept the problem is this still has a function in it and so we take it that step further to get our exact point you