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4-19 AIM3 8.8 Laws of Logarithms

Added Apr 19, 2024 •

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okay today we're going to see how we can use these laws of logs to help us in certain situations I'm also going to just buy the change of Base rule that we met yesterday and let's just get to it right so honestly I'm just reminding you here on this page of some of the properties of exponents that we discussed before you need to pause this page to remind yourself you should do so remember the most remember the great relationship between logarithms and Donuts all right so here are fundamental laws of logarithms log of some legal base work also important that should hire like that restrict on The BasedGod be positive but I can't equal one and so what do we have for the log of a product is the sum of the logs the log of a quotient is the difference of the logs the log of something to a power is that excellent that power times the log know why do we have these well they're only around because they're incredibly helpful so for example what used to happen nowadays in your calculator but but I'd like you to see trying to understand a little bit of what's what's going on with the great utility of locks so people were able to discover that the log of 2.4 was approximately .38 you could do that jump on your calculator do 10 to the point 380 it's about 2.4 well that's fine then suppose you wonder the log of 24 you say I will always like I do a new calculation but here's the beauty of it because of our laws of logs you don't have to move calculation so check this out another name 424 Corsa B 10 * 2.4 why is 2.4 so interesting already know what it is we also know that the log of a product take it back up here tracing but when you log of a product is the sum of the log so that tells us that we could rename 10th the log of 10 * 2.4 is the log of 10 plus the log of 2.4 but wait log of 10 that's easy log of 2.4 is already known so log of 10 is one lot of 2.4 we've discovered is about .38 and put it together and hopefully you can get 1.38 she can see once we knew 2.4 we really have access to a lot of other values and I bet now you can figure out the log of 240 without too much effort if you think of it as see if I can write machine colors you think was the log I've a hundred * 2.4 and we went through the same process here at you can you can see we can end up with a .38 forgiveness will baking show you I'm probably helpful Discovery or the invention of logarithms were greatly simplified life so many people working with numbers I felt the blood see why these are true then we said their true u-prep seen before but let's take a look so what do we have here let's say be the extra clothes and be the Y close in and if we didn't see that that's right applying the exponent and we have be the x x b to the Y that must be the same as a x in since. EXT four sensors dad and I'm so fine no big deal let's build on that let's see why this particular the first while we met is true so by definition be the x equals am then log base B of M equals x are translation from the exponential to log rhythmic in the same token be the wife will send Log please be of any gold watch that makes sense you let's move on okay so what does that tell us well then the log of MN must be the log up well since animals be the accident be the one I'm going to replace and would be the action and with B to the Y that's fine and then since I'm multiplying the same basement so I can add exponents and here I thought I just remind you something that we've talked about in the past but let's just refresh our brains here a certain property we have log base B of B to the all right we got some bass we got that face raise to some power then can you predict what are log is going to be another words be to the what is be there probably and this is this is a nice little property and that if we can set up in a situation where this face and this face with the same than this exponent much impact be our log and why is that helpful because that's what we're doing right log base B and B to the X + Y So based on what we just talked about there that must mean but I have x + y and I know what X and Y are I had that way back here x is log base B of A why is August the event so now I can rewrite this by substituting back and look where we started we start the log of a product now we have the some of those lost so now we've proven that particular rules logarithms you could use a similar approach to prove the other two you might have to practice that come the next slide we're going to see how we can use logs to help us solve a particular type of problem so let's see what's going on yet it looks I meant to keep that confront hope you didn't notice so Ariella puts $500 encounter it's 9% interest per year compounded annually okay how many years will take to double so hopefully remember our compound interest formula which I actually left for there so the principal 1 + the annual rate divided by however many times is compounded in a year times a number of compound periods to how long now in this case of course I'm just see here question is how many years so that keeps working me solving 40 let's go out well our balance how long will it take to double start a 500 choke how long till we get to 1000 we're putting 500 in at 9% Seuss's how we start thousand dollars is 501.09 to the tea and then how long should we could just try on that would be very interesting let's just do a little a little math here so since we're trying to solve 40 oz Lee one isolate that term so I reckon we should / 500 and we get that now you might recognize Justice is kind of nice situation we got a nice exponential equation here how do we undo exponential's when your unknown is in the exponent like you want to get it out of there I really great tool now we different suppose this was too and this was for when we could rewrite 4 as a power to make and use them the fact they have the same base but in this case it's not true but we can a more standard tricky will do more offense in spaces aren't usually going to be able to be written in terms of each other as we ever will. Let's just take unknowns out of exponents if it comes we have laws and of course this was an equation I can do the same operation to both sides, take the log of both sides which means I would get log2 and then one of our rules remember backup is if you have a log of some antilog to a power I can bring that power forward as the multiplier use again you might want to see if you can prove that using the same kind of approach that you so that means we can call this you have this t log 1.09 an hour business cuz work member we're trying to solve 40 + this divided are the logs 1.09 am I picking base 10 on a calculator and it would turn out to be about 8.04 segesta shade over 8 years will take for this $500 to grow to $1,000 chiffon but I want just look at it different way just to show your different approaches and neatly should be familiar with that again we have a thousand dollars equals 500 * 1.09 to the T what if we take a login okay no we can actually meet you in these numbers are kind of convenient so what's log of 1,000 what's log of something times something turns out log the Thousand is just 3 and the log of a product is the sum of the log so I can break this apart like so just get this out of the way so now what you can do please just do some some math here so we could say three- log 500 equals and the word why did that happen I was nowhere near the screen got a little bit nicer I hope see what happens here does and then here's our log of a power so we can have tea time long 1 points doing it again isn't 1.09 and since we're trying to solve 40 / long 1.09 and jump on our calculator and get whatever we get in just a little aside I would just point out something to you log 500 keep in mind how we can rename things well spoke to her to call it log 100 * 5 well now log 100 plus I'm on now plus Log 5 and a log 100 so convenient 2 + log. I ties into at that conversation we were having with 2.4 and 24 and so on and remind you that little property so we see two different ways of using logs to solve a I can think of on the left is a little bit cleaner but you know sometimes one of the right might be more helpful by next I just want to remind you open else I want to go back on something real quick it's 72 here you know I'm a big and of is my answer reasonable can I can sense of what my answer is before I I get it turns out there's this rule called the rule of 72 that works when you're compounding compound interest and you're certain how long it takes to double long time about Dublin question so it's a specific rule the basically says that take your annual interest rate 9% and divide 72 by it so 72 / 9 is 8 remember answers just do great this is just a quick way of estimating the how long it will take in fact let me just call this and you can get the URL there and if you want to Regal that more about it you could just a nice little bit just have to continuously compounded compounded and it just discussing I'm not going to school then you can always come back and look at this I've got to wait for my screen to get ready for me again as soon as I can get there we go all right so again just a reminder of various properties we talked about you won't be familiar with these might want to make flash cards and fill out the left side and put the right side on the on the other side you know how do you flash cards last night change of base law that we said for example I had to log base six of 100 I want to pick a hundred cuz that's too nice and number well sweetest Gus we could call this log any base I watch I'll go, show log 101 divided by log why that's what I want to show you now so long jump baseball blah blah all right so by definition go back to the school by the way this must be next to the a equals y i well remember I'm trying to figure out what a is right back again I got a up in an exponent how do I get unknowns out of exponents we did it a few minutes ago we take logs if I take the log of x to the A of course I would get a log base B of X and I hope you have this and then now I'm just trying to stall for a so we would simply divided by log base 3 of X till we end up with this right here and there's a rule y over X like like you might recall 1016 log 101 / log 6 so there's the proof for it while we can use it and it really is pretty handy daycare

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4-19 AIM3 8.8 Laws of Logarithms

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