Stony Brook MAT 123 Fall 2015
Lecture 16: Compound interest, e, and the natural logarithm
October 19, 2015

Start   so theres a lot of stuff we should go over, exponential stuff we should go over so remember last class we talked about what a logarithm is so a log has a base and that tells you what power you are raising the logarithm to the logarithm is just an exponent
0:31log base b of x equals a means you take b and you raise it to the a you get x so much easier with a calculator but you guys learn about ph in chemisrty, its an acid ph is a log scale ph7 has
1:016 has 10 times as many ions than ph7 ph 5 has 10 times more then ph6 thats a lot of ions okay so you see logarithms you use logarithms often on scaling because otherwise your numbers are so big and spreading so big you cant get a nice graph but if you do the log of a number
1:36log base b of x equals a, so if you had something like, log base 2 1/5 (unintelligible) and i ask you a question like that want to figure out what that was so we would say that equals something so well call that x doesnt really matter and that tells me that 2 to the x
2:001 over 500 log how am i suppose to know what 500 log is, a good guess would be the power of 2 without a calculator thats just b so you go over here and start testing things and start writing powers like all the numbers maybe not quiet ((unintelligible) but thats okay and you do 1 2 3 4 5 6 7 8 9 (unintelligible) 1/2 to the 9th
2:34so were not gonna do 65 5 36 because thats to big, you should guess like its 2 or something or 3 something, yes?
now we're not done 2x, 1/2 to the 9 has to be negative -9 thats what youre looking for right?
that make sense? you should be able to do something like that
3:21okay i think i remember seeing something similar to that doesnt really matter i can make up a different number
3:39how bout that?
what if i gave you something like that well again you just use this rule, this definition that tells you that 4 raised to the x give you 12.6 i dont know what 2 to the 12.6 is thats okay youre not suppose to know what the 2nthe 12.6 is
4:02you just have to figure out what x is this 4 is 2 squared that tells me that now i can multiply the powers, 2 to the x has to be the same as 2 times again, 2 to the 2x has to be the same a 2 to the 12.6
4:322x has to equal 12.6 x has to be 0.3 thats another type of logarithm youre gonna have to do s whats going on? were playing the bases and more through the exponents these arent so bad, we learned logarithms in high school some of you?
5:00we okay? from back in high school?
so back to understanding these things a bit so logarithms show up in chemistry and physical science so for example, earthqauke and wicker scale wicker scale was a logarithm scale again it works in powers of 10 so if you have an earthquake thats 3 in wickle scale might even feel it, you might feel a 3, we had one here a couple years ago
5:34yu just kinda wobble a bit 4 is times as much energy as a 3, its not 1 more its 10 times more so an earthquake at 4 you would feel might not knock you down but it might an earthquake of 5 is 10 times more an earthquake of 4 an earthquake at 5 youre gonna feel it, its gonna its gonna knock you over, its gonna make things rattle 6 is 10 times more than 5 so its a thousand of energy then 3
6:04and earthquake of 6 can cause a lot of problems you wouldnt want to be driving across a bridge 7, 8 theses things are catastrophic we had an 8 earthquake in the past you were all old enough to remmeber that right?
even from 3 to the 8 thats 5 orders of magnitude
6:3110 to the 5, 10 times 10 times 10 times 10 times 10 100 thousand times more energy thats a lot of energy decimal scale is a logarithm scale so 7 decimals (unintelligible) thats pretty loud if youre up close to those speakers thats somewhere between 90 and 100 decimals at about 120
7:00youre an airplane jet engine at 120 decimals, so even if its only 20 decimals more, its so loud youre ears are all messed up from it or if you take your ipod and you stick those buds in your ear to the highest possible volume to hear whats going on thats close to 120 decimals so if you were your ipod with your earbuds and you crank that up
7:30your going to be deaf, its going to rip your eardrums youll fix it, youll have something electronic stem cells or maybe youll be deaf and go to (unintelligible) i recommend that as a career thats because its the energy you are measure, the earbuds are this close from the eardrum thats why you should where the over the head earphones not earbuds, so close youre absorbing an enormous amount of energy
8:00well who would turn them up all the way boze or beats or something like that so when youre in your car and you have that thing were the car next to you is vibrating a lot of decimals, so the numbers are a very small difference so youre talking the multiples okay that represent by same changes in number and thats what logarithms are for so ph so acid so a ph4 acid is strong i think
8:31i think lemon juice might be more than 4 you certainly wouldnt want a ph3 spilt on you 7 is normal, again just small differences so that could be 5 or 4 then you try and graph it you cant scale it on the same piece of paper its very hard to have a 3 and 30 thousand on the same scale logarithms make these huge curves, into striaght lines so thats why we do logarithms
9:02have we learned about our earbuds we use them anyways arent we?
thats why people smoke ill worry about that when im old im not old right?
good answer yes, youd be in trouble how about we said the other wayb of a logarithm
9:41what if i said log base 6 x equals 3, now i have to find what x is (unintelligible)
10:006 to the 3 equals x if this were a test you wouldnt have to find what 6 to the 3 is, just write 6 to the 3 if you figure it out the problem with multiplying it out is you could get it wrong you should 216, but if you write some other number it displays you cant multiply 6 times 6 times 6 thats not a good idea you could write i dont know what it is, what is 6 times 6 times 6, or you can do 6 cubed, thats fine with us okay?
10:31so far so good?
other types of stuff oh yea we had a couple of these same exact questions ill do those in a little bit, cause theres some fun stuff you can do with that
11:01look at a question like that how do we answer something like that? well use the definition its says b to the 4/3 is 625 i have no idea what number to the 4/3 is 625
11:32what you can do is say well thats just 625 to the 3/4 and thats your answer okay you can figure out what 625 to 3/4 as well, that means the 4th root of 625 cubed okay? yes?
12:00you should be able to put this in webassign, it should except that this one, make sure you out that into webassign you can certainly put that one, yes you could put either one if you do the (unintelligible) correctly because you can do (unintelligible) to the 4th root
12:33so you may not have put the (unintelligible) n the right way so thats tricky cause you have to have two different radicals in webassign theres one thats just a square root, and theres one with a box that you put 3 4 or 5th root i believe you i can check it later yes you think you could what?
13:04cubed root of 5 well not for this problem oh it could be yea i have to look at what you individually put in i have a way to see not only what you put in correctly but what you put in incorrectly before you got it correct and all that so i have to check, however theres many of you in this room i cant even count that high, its bigger then 10
13:30so i would have to sit there for a long period of time to look at your answers i have very little life but i have that much lif so right when you go from this side to the other side yes you turn the radical upside down so what you do is cube both sides and you take the 4th root of both sides if this is a over b then this side is b/a
14:03so the left side if you have if you have x to the a/b equals y thats the same as saying x and y is b/a that make sense?
i see a lot of sleepy faces today
14:33okay so what is the 4th root of 625, what do you think?
5 is a good gues its gotta be something you can figure out without to much work because in theory use can do without a calculator 5 cubed is 125, so this is 125 powers of 5 that are greater than 1 end in 25
15:00just in case you were wondering so 5 to the anything, anything bigger than 1 will end in 25 25, 125 625, 3125, 15625 and so on, okay?
so they keep ending in 25 so thats a clue okay powers are as even, you think you can figure out why?
what else could we give you? oh right other stuff you can do with logs so we were talking about logs in exponentials
15:32the other thing is the log exponential are inverses of each other
16:40alright so how do we do that? well we write that its y equals log base 3 to the x and you want to find the inverse, right so switch x and y and then you get 3 to the x equals y and thats the inverse so if f inverse of x
17:01is just 3x to the x thats the key, logs and exponentials are inverses of each other if this goes to the point 9,2 this goes through point 2,9 because theyre inverses check it out and see now one of you will raise your hand and say thats what i put in the webassign and it marked me wrong thats because webassign ask for things like f inverse of y
17:30f inverse of y, what do you think f inverse of y is?
or look at this equal to y is that what you were gonna ask i know so pay attention when you are doing webassign thats usually what i get
18:00i dont know why webassign does that but it does if it ask for f inverse of p then you will use p doesnt matter what the letter is, in fact you really want to hurt people on the exam you could put in pi any time you want to lower grades you put in pi how bout something slighly worse?
you have this written down?
18:33lets have you do a practice one that should keep everybody busy for a few minutes (unintelligible)
19:11so switch x and y thats our first step thats switch x and y but maybe we actual do the inverse now you need to isolate it to write in the function as soon as you switch the variables you reflect across the line y=x now everything you do is coming out converted
19:34thats really ball so then you say yes yes but thats not enough, you want to have y by its self so 4 to the x is 2y-1 add the one to both sides you get 4 to the x plus 1 equals 2y so 4 to the x plus 1 all over 2
20:01y and thats the f inverse thats not so bad how do you feel about that one?
so this is yet again, switch x and y now you have x is log base 4 of 2y-1 so to get rid of that log base 4, you move 4 to both sides. exponentiated both sides now we have 4 to the x y-1 and now you have to isolate log you add 1 to both sides and divide by 2
20:33so thats the inverse of x what if i go the other way what if i go the other way? what if i have
21:09how would i do that one?
i would write that as y=3 to the x+1 switch x and y now what?
21:30you take the log, (unintelligible) perfect so you actually have 2 options here okay one you can take any log you want, i will show you that in a second or you could do log base 3 so you do the log base 3 of x is y+1 so you subtract 1 from both sides you get y is log base 3 of x minus 1
22:04you dont have to practice this method, but be careful it doesnt look like youre doing log of x-1 okay dont get those mixed up so far so good?
the other thing you could of done instead so when youre at this stage is you take the log of both sides any log you want
22:33now with the log rule you can put the y+1 in front you get log of x plus y+1 log 3 see what i did? you bring the power infront and now divide that by log 3 and get log x over log 3 is y+1 and subtract 1 from both sides
23:01that should tell you something, that should tell you that log of x over log of 3 is the same thing as log base 3 of x in fact it is and some people change the face rule im going to slide this up for a second so the log base b of x is the same thing as log, some other base over log b
23:30and usually you use log base 10 or some other type of log okay?
you learn another type in a minute thats called the change in base law change of base rule change of base rule says, if you have a log base b of x
24:01you just make it log of x over log of b so for example if you have log base 4 of 11 thats lo 11 over log of 4 so many of you have calculators where you cant find log base 4 over 11 but you can do this so thats the way you do logs over bases if you have t84+ it has log of separate bases built in but with log t83 does not
24:31got that?
thats why log base 3 of x is also log of x over llog of 3 what other type of log thing so we use logs to solve exponential equations
25:12so how many of you are in the programs in the business school? lots of you?
some of you gotta go to business school?
gonna teach you the key to get an mba in finance , ill give you guys a finance eduction right now you dont need to go to business school, you need to know one equation
25:30compound interest, anyone know what compound interest is?
you might have saw this in high school maybe compound interest, wait till you go to business school youll be using the same formula over and over and say come on guys give me another formula compound interest formula says sometimes we call that perks how much you have in the future depends on home much you have now you take it and multiply it by 1 times the interest rate
26:03raised t t is time so r interest rate t is time p is what you have present or also known as present value and f is how much you have in the future also known as future value
26:34i use to have to do this stuff for a living thats less fun wall street is a little less fun then you think but you make lots of money thats okay it balances it out, thats okay it compensates the functions so for example
27:06this stuff is all built into calculators these days so next semester you call your parents up and say gosh books are really expensive
27:32i may need a thousand dollars for textbooks this semester so your parents elieve you and give you a check for a thousand dollars and you can go a download all the pdfs from some website cause i heard thats possible and you can pocket that 1000 dollars and you find a bank thats crazy enough to pay you 6% interest right now youll get .6 but right now well say 6 percent
28:02and you forget you put the money in there and then one day about five years from now you go oh wait i remember that thousand dollars i took from my parents i wonder what it is now because that thousand dollars can be worth a lot of money in 5 years
28:306%, youre lucky if a bank gives you .6 good luck with that because a thousand dollars theyll laugh at you but if it was a billion dollars, start to be real money okay so you just use the formula i guess i could show you guys later where the formula comes from, its kinda fun you put a thousand dollars in its going to grow 6% for 5 years and then i dont know what that is
29:06whip out a calculator oh i was close .23
29:321338.23 thats not bad for 5 years keep that your whole life and youl retire with lots of money thats a target number the trick is to get 6% for the last 7 years the historical interest has been 1% we need to live 5 or 6 life times to get the money .1. well actually your bank right now
30:02checking account .01 percent so it will take you about 7,000 years to recover your money i dont know about you but i dont plan on taking this class 1,00 years from now i know that it feels like it last that long but it doesnt actually other things you can do with your bank accoutn
30:30now banks are [unintelligible] and they say why would we bother going to bank number 1 that pays you 6% percent a year, we'll pay you 3% percent twice a year instead another bank says well pay you 1.5% four times a year thats called a compound [unintelligible], so you take the formula and modify it a bit the end is the number of times a year they will provide you interest
31:04so lets accounts normally pay quarterly you leave the money in and at the end of 3 months theyll give you interest so 6 percent thats for the year so once a quarter theyll pay you a fourth of 6 percent however you earn interest for whatever you leave in the bank so if you need the interest in the bank you will gain compound on the interest so addition to what you started with
31:31you will now get interest based on your interest thats called compound whats nice is 3% twice a year is better than 6% once a year so lets say we had a 1000 dollars and we had 6% twice a year, semi annually so now were gonna get it 10 times, instead 5 times because youre gonna have 5 years and you get paid twice a year
32:01so this will be a thousand times 1.03 times 10, instead of 1.06 to the 5 and that comes out better now you get 1343 dollars and 92 cents thats better thn the other bank, that bank was only
32:351338.23 so yea 5 dollars but remember the big people are paying billions and billions of dollars so that makes a big difference even if you had 10 thousand dollars thats a big difference you should say to your parents you should start stocking now your gonna win a lot of that when the time comes but 3% twice a year is better then 6% once a year
33:00how bout paid monthly?
if we paid you monthly you get 12 times a year and you get 60 rounds of interest
33:30that is 1348 dollars and 85 cents. thats a lot better now youre starting to get some serious money here at 10 dollars better i owe you a thousand for a million dollars thats 10000 dollars better 10,000 dollars can be downpayment for range rover drive it and someone will come take it away, and then it will be dirty i mean youre not gonna take the whole thing
34:02so notice the more frequently compounds the more money you make a smaller number more often is better then a bigger number less often so now you say this bank is better then that bank now you go to another bank that says were going to pay you infinitely often so they say, were gonna take that formula and were gonna take that r and were gonna divide it by infinity essentially a trillion
34:32doesnt matter, and were gonna pay it all the time so we always pay compound, so thats continuos compounding so if something compound continuously its a little tricky, you guys havent learned calculus yet basically you are taking r and your sort of dividing it by infinity and this is infinitely often so that doesnt make any since, so a mathematician worked on this for a while and they discovered
35:02that you got si e is the number when you take r actually you get e to the r but if you take 1+1/n and you raise that to the n and you make n bigger and bigger and bigger
35:31as n goes through infinity it starts to look like number e e is about 2.718 e is one of those special numbers like pi it just shows up all over the place lots in finance also in physics engineering, biology so biology you have the cells and theyre always dividing, theyre always changing they are actually discrete you cant really have an infinity test often infinity test can be often its gonna be small, but at some point
36:03at some point it starts to look like e, in fact at like 100 times, it starts to its just a compound daily its almost the same as compound continuously when you look back, by the way, you get the interest rate they would bore you on continuous compounds but in the old days before you got smarter computers and calculators they can take the advantage of you by offering 1 rate
36:31verses another rate okay then a bank supports the difference so this is e so e go to your calculator stands for 2.718 and you got a lot of fun with e now you invest a thousand dollars
37:02at 6% compound and continuously, all the time now how much will you have, now youll actually have
37:511349.86 thats not a lot better than monthly but thats a little betetr its still better cant do better then continuos compound thats as good as it gets
38:03when banks told you what apr annual percent rate they are giving you the equivalent of compound and continuous rate what do you need to know e?
well e is actually what you use for all the logarithm stuff except for logs that use base 10 pretty much when you use logarithms in math almost always base e
38:31because you want to know e of something and that is everyones favorite log the natural log also known as mhm pronounced log but other people pronounce it silly your a math teacher in high school probably it stands for log (French) so remember our definition before logs
39:01so remember you have log base b of x equals a means b to the a equals x so now if you have natural log of x equals a that means e to the x equals a because ln is really log base b
39:32b to the a yes sorry good thing i brought you i know it doesnt really work that good so dont start that was very good ill give you the chalk,no? you sure?
so natural log, you can do all rules for regular common logs, other logs are true for natural logs
40:04nope im gonna rewrite tha natural lo of 1 equals 0 just like it did with any other case
40:30because its base e thats because e to the 0 is 1 the natural log of e equals 1 natural log of e to the x equals x if i had the natural log of 2 things multiplied together
41:00the natural log of the first plus natural log of the 2nd just like the other logs, okay?
so get use to seeing this ln, cause youre gonna see it in calculus pretty much in calculus and beyond, when you hear the word log were taking about natural log you mean oh you meant ln, oh yea natural log if you had the natural log of a to b whoops got ahead of myself there a/b
41:31thats natural log of a minus natural log of b and if you have natural log of a raised to the b b times the natural log of a, all the same rules as before
43:06lets do some playing with natural logs for a couple minutes
43:31how bout that?
you ready for this one?
so we write this as y equals natural log 2x-5 and switch x and y you get x is natural log 2y-5 now what do we do we do e to both sides you get e to the x
44:01is 2y minus 5 thats cause its log base e so now you add 5 and divide by 2 and that is your inverse function see e to the x is very special
44:33e to the x is a function where in calculus you learn how functions change the way it changes is equal to what its to itselfs i dont know how to describe that but essentially at any moment the slope of the curve [unintelligible] very odd why do we add 5 well cause i have 2y-5
45:01so i want to isolate y so i add 5 to both sides divide by 2 were you guys able to do that one?
yea lets try a different one can i erase yet? no oh why do i have the e?
remember natural log is really like saying log base e
45:30so if i want to get rid of the log and exponentiate both sides, so e to the x equals this lets try another one of these
46:00okay lets find the inverse of that one
47:57how do we do this one
48:01you write y equals e to the 1+6x so switch x and y, its always the second thing okay thats the easy part now what do we do?
take the natural log of x and that equals 1+6y cause remember natural log and e are inverse functions
48:35write out the base e so we have x to e 1+6y and the natural log of x is 1+6y so subtract 1 and divide by 6, so element of x minus 1 over 6 equals y
49:01so thats the inverse function, so webassign said f inverse of y, you would make it the natural log of y minus 1 over 6 no that ln well cause remember natural log of x means log base e of x so ln is just a short way of writing log base e
49:31it stands for log natural in french and its just one less letter or 2 less letters actuualy so that pisition we are lazy but were just trying to find compact ways to write that lets do one more of these and make sure we can do it
50:21theres one more to entertain you
50:35so first you start off with y=e to the 4x-3 all over plus 2 and we switch x and y so far so good, everybody go that step?
thats worth something okay subtract 2 from both sides and now take the natural log of both sides
51:02you get natural log of x minus 2 equals 4y minus 3 now you cannot combined this 3 with that x minus 2 thats the log of x-2 so if the log of x-2 plus 3 equals 4y or you can write 3+the natural log of x minus 2 you can take the x-2 and add 3
51:31its not x+1 its x-2 plus 3 now divide by 4 that is inverse function got the idea?
yea you should use those perentesis you dont want to confuse yourself for us webassign, you can write it as 3ln of x-2
52:00dont want the x-2 in front 3 plus sorry lets practice one more of these, i wanna make sure you guys can do this i wont erase
52:50why did i take away the 4?
where here?
oh i was just saying like this and this
53:01your whole world was shaking there for a second alright are you ready? could i erase?
53:53how bout that one
54:19cmon guys you can do this one
54:32cant use the inverting cant use the e in there until you got log by itself but first you switch the letters okay now youre gonna do that whole e thing, but you only do that until you got just natural log on the right side so first subtract one from both sides
55:05divide by 2 now you can do e to both sides now you get e to the x over 1-2 is y over 5 now im just gonna divide 5 by both sides
55:33i get 5 e to the x minus 1 over 2 y and that is the inverse howd we do on that one yes loving life not so much
56:06theres a variety of ways to get to this step