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
log 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
6 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
log 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
1 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
so 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
okay i think i remember seeing something similar to that
doesnt really matter
i can make up a different number
how 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
you 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
2x 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?
we 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
yu 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
and 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
10 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
youre 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
your 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
well 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
i 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
have 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
what if i said log base 6 x equals 3, now i have to find what x is
(unintelligible)
6 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?
so 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
look 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
what 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?
you 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
so 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?
cubed 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
so 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
so 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
okay 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
just 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
the other thing is
the log exponential are inverses of each other
alright 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
is 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
f 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
i 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?
lets have you do a practice one
that should keep everybody busy for a few minutes
(unintelligible)
so 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
thats 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
y 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
so thats the inverse of x
what if i go the other way
what if i go the other way? what if i have
how would i do that one?
i would write that as y=3 to the x+1
switch x and y
now what?
you 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
you 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
now 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
that 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
and 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
you 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
got 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
so 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
compound 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
raised 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
i 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
this stuff is all built into calculators these days
so next semester you call your parents up and say gosh books are really expensive
i 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
and 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
6%, 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
whip out a calculator
oh i was close
.23
1338.23 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
checking 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
now 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
so 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
you 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
so 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
1338.23
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
how bout paid monthly?
if we paid you monthly
you get 12 times a year
and you get 60 rounds of interest
that 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
so 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
doesnt 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
that 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
as 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
at 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
verses 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
at 6%
compound and continuously, all the time
now how much will you have, now youll actually have
1349.86 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
when 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
because 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
so 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
b 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
nope im gonna rewrite tha
natural lo of 1
equals 0
just like it did with any other case
because 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
the 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
thats 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
lets do some playing with natural logs for a couple minutes
how 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
is 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
e 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
so 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
so 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
okay lets find the inverse of that one
how do we do this one
you 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
write 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
so 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
it 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
theres one more to entertain you
so 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
you 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
its 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
dont 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
why did i take away the 4?
where here?
oh i was just saying like this and this
your whole world was shaking there for a second
alright are you ready? could i erase?
how bout that one
cmon guys you can do this one
cant 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
divide 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
i 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
theres a variety of ways to get to this step