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4-18 AIM3 8.7 Defining Logarithms

Added Apr 18, 2024 •

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alright the time has come for us to formalize something we've been building up to and mainly that is our friend the logarithmic function here you can see the definition this reminder we've been calling it The L function so we discovered the exponential function has an inverse that is an exponential function which is a one-to-one function has an inverse we work on the L function now we know the invert have an exponential is a log function there's some very important consequences of dip a little bit about this notation you see here whatever reason the author said they would change the notation on us so this I'm right here it really is no way saying that why equals log base B of x I chose to do that now I don't know and then this is just the exponential function y equals B to the X and these are inverses of each other and we discuss now falling Point really important if you can appreciate understand it then this will make your life much easier it's Q relationship between exponential and logarithmic functions if you have an exponential function here we go be the k equals in well we know this relationship from exponential and logarithmic that means I could take be the cake was and I could rewrite it as log face B equals 10 the way we translate log to exponential exponential log this value here raise to the k equals a feed to the cake was that the effect may call me to the king this is just so important and of course Scituate Street if you have a log equation you can change it exponential if it's more convenient now I would also like you to clubs or something here log base B of M equals to k equals k k equals this there's an expression you can hear lots of people are with me but many people say mathematics in here we go a log is an exponent if you're going with a log that you know it's it's the exponent in some particular exponential equation and this ability to bounce back and forth is very important I'm in geometric Lee well how did the grass compare if you were to graph an exponential function graph its accompanying its Envy how to craft a well I hope you said that after all they are inverses of each other so let's start playing around just a little bit something I miss and previously is that if we don't draw a bass drums call the common logarithm the base of 10 and that's when we played in a little bit on our calculators and may have another class I pop quiz why is it some people say log 9 to the base trees some people say log base 3 of 9 I'm not going to work either way why is it correct to say this well if we use our definition what you just said and this must mean r e the base actually I'm just recently out give that to you in a second this must mean three squared equals 9 and that's obviously true statement vocab list down here course is the base we just chatted that this is the log this number here called but you get it antilog accept the Cavaliers always very important around these things okay so I want to solve this particular equation will just say the answer some number n Nick something about the answer before actually talking can you tell me anything at all about the value of n based on our exponent rules are definition here rather you know this 5 to the end equals 1 over 25 can you tell me anything about him 57 equals 1 over well I hope you are thinking of oh well that must mean that and difficulty in and mo do negative yes yes indeed that must be true so it's nice to know something about it further more if you think about you got a fight here in a 25 here. Might give you some idea about what it must be so now we know but then negative that's nice and snow my also misses something is going to be negative to find the number of the convenience of course go for it might also observed you can say 5 to the end and then the 1/25 we could rewrite also is a power of five of course it would be 5 to the negative 2 until we get and is -2 be able to rewrite them with the bass the same as very convenient sometimes not possible we'll talk about how to deal with that cable what about this problem over here can you tell me anything about what the solution would be nothing, so yeah I guess I'll call it something else says call it ready so 14 it's working so well a minute ago 1 4 to the A peoples 16 since we're going from a 100 to something that's not one over I hope that's a big clue you that he's going to have to be negative I have this thing is just so sensitive and again you might notice that 4 and 16 both me written in the same base so we could call 1/4 4 to the negative 1 that is raised to be a and then Bruce gets 16 Maybe of course it's 16 4 squared plus nicer the same base we get four to the negative a equals 4 squared and if I can squeeze it in - A is 2 and so a would have to equal -2 send back a little bit of mess you there let's look at a couple other situations still got these different logs and noticed that these are not all common logs this is a bass 5 this is a Comm log base 10 but don't have to ride 4 Bay Street that's fine again I would like to think of reasonableness was take this first problem log base 5 26 what's you can come up with an answer for that the possible proximate if we have to well by definition of course this equals something so using my definition or my trans from log to exponential perhaps I cannot read write this as 5 times something equals 26 and I would hope that after a moment's reflection you can see that the end well 5 squared is 25 so we got to be bigger than 2 course 5 cubed is 125 so must be I know why it does this 5 q is 125 Sanchez got to be between 2in 3in ABS is much closer to 2 than it is to 3 since 26 years are close to 25 more about that I think on the next slide or the one after I think it's the next one at this very moment we could just make a proximation so we can stay well let me just do a little bit bigger than 5 and I might trying now I'm sorry little bit bigger than 24 might try 52 double-wide or no 2.1 do with that does I know we will go out to two places that's what we should do and I've discovered that 5 to 2.1 is too big or too small and then I would adjust it accordingly and in time I could Discover at least two places that five to the 2.0 2 is about 25 hate to sew in the neighborhood of 26 I want to be more accurate I would go out to more decimal places I just said I'll give you another method of that in a moment be long and I don't have to take B base-10 by the you don't say anything is understood to be based a just a log of 1/10 Oyen using our definition if this equals something I think I may have set it up already let me know what is Lexi what's on cover it Japan okay. So by definition tend to the end must be equal 1/10 I would hope he's being powers of ten just jump out at you in fact 1/10 to be called 10 to the negative one but that's very nice hand something equals 10 to the negative one will then does something obviously is 1 how about sea well from the same Reckoning we know that hurry up that must be forward to the something equals 32 now I suppose you could just try guessing cuz we know for squared is 16 and 4 q to 64 is okay so it's between 2 and 3 but there's something that's facts about this is similar to what we saw in be might observe that both 4 and 32 can be written as powers of 2 so for example I could call for to the end 2 squared today at 10:32 is 2 to the fifth exponent lossless B2 to the to end 3 + 2 to the 5th and of course this makes us very happy cuz the bases are the same now we can simply say that to n equals five you so end must be a 5/2 or 2.5 soundbar to work out to be exactly correct we love it when we can make the bases the same bases can be made the same way we go life's very easy sometimes they can as I said I'll show you a technique for doing that well how about Part D how would you rewrite that well of course maybe three of the antifa -1 and we can see I hope we remember oh that's not possible 32 anything but can be a positive number I'm sleep it's to a positive value you get 50 you get one if it's negative you get 10 and so this simply has no wolf Issaquah play keno solution so the ability to change from exponential to log rhythmic it is very important something I'm already mentioned mentioned that I was going to mention really how can we find a box mix without guessing check that is a here I said well we know it's bigger than to let's start guest until we get close enough it will only bring it to two decimal places well okay but be nice if there's a better method guess what now people who got a better map it's called the change of Base rule allows you to change faces so what it says is if you have some log illegally to find of course then you can rewrite it look what it says look see how the base exchange has gone from Basics to base B can be some legal bass although for our purposes at least right now we will typically make it a common law because base-10 is right there are calculated life easy to notice you can change log base acts of Y into log base b u r y over Long Day's events just y over X sort of like their looks so forgetful here one we've already done log base 5 of 26 points definition I could rename that anticipate what you going to be log 26 / Log 5 jump on my calculator and put it and I would discover as we've already discussed it's about 2.02 when we get trying to get that that's a little bit better Miss changing baseball is super handy if you're not I'm with a base ten situation which is convenient or in situations where you if you can make the bases the same great but if you can't make the bases the same then the change of Base formula any very handy and here I just copied a summary of various properties of log functions should be to these make sure you understand why each of these is correct you can tell something about the intercepts of a log function I when it's increasing one's decreasing and all this last slide I just have a picture of your standard this is common log base 10 base ten ten to the zero would be one so that's what we have here we raise 10 20 we get one when X is one maybe I can work this out when why is 0 X1 we have 2.10 there it is when x's on shooting when why is one we we get 10 to the first answer the point 10 one in there it is so here's our locker it approaches the the y-axis asymptotically never hit just goes down Trevor depending on what the value you put in between zero and one so we have formalized logs a little bit will be playing with them take care

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4-18 AIM3 8.7 Defining Logarithms

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