The most important thing you need to remember
from this video about applications of logarithms
is basically... Anytime you have a variable
in an exponent, the only way you're going to be able to solve for it
is using logarithms. So that's the power of logarithms; solving for variable
that's up in the exponent when you really can't reach it.
and let me give you a scenario where we're going to use it
consider the general interest formula
for multiple compound in this involves your money at the bank
a stance for the accrued amount that you get the total amount accrued principal
an interest over the years if you've invested
p amount
love principal okay that's the amount you put in originally
r is the interest rate
he is the amount of time that you invested
in years and and as the number of periods per year for instance scored
early
and would be four or monthly
and would be twelve except for now before you get nervous
but the answer a question using this formula without using logarithms
consider how much
would you the investor accrue how much money would be
in your account if you invested five hundred dollars and interest rate of 7
percent
for four years you kept in there for four years when the in
interest is yet as typically it is compound it monthly
well let's put the numbers in
our principal is 500 as you can see
.0 77 percent for the interest
and the magic number for monthly is 12 and the time
key if you would is four
now all we have to do is use this formula basically
I would put this into a calculator make no mistake
k cuz there's no solving and this is what you have to type please note:
that you have to put the exponent
12 times for in its own set of parentheses
anytime you have an exponent has to be in parentheses
okay be careful with parentheses
or you get an answer it just won't be right me answer in this case is quite
logical after four years
five hundred dollars will have turned into 661 dollars and three cents
let me do this one more time just to make sure you're familiar with this
formula
okay let's start off again how much would an investor that be you
accrue how much were they have in their bank account
throwing down three thousand dollars and interest rate
have 10 percent at a hefty interest rate but for five years
so it's in the bank for five years when the interest is compound did
quarterly on that means a magic number is four
so are putting in all our numbers we're gonna have a $3000 for principal
10 percent or point 104
interest rate and four for the
and and of course five for the time in years and if we put this
in do our into that calculator
it should basically look like this once again I warn you make sure you put
the exponent in parentheses okay
and you got a lot of free money they remember you started off with three
thousand know you had a high interest rate you waited five years
but if you think about it your three thousand turned in the 4900 something so
basically that nineteen hundred dollars a free money
that's what I call it anyway okay so that's how this formula works
okay let's get back to what we can use the logarithms for
well let's look at the same formula but we're kinda gonna ask a different
question
here we go how long now we're asking for time
how long would it take you an investor to double your investment
six hundred dollars if you had an interest rate of 12 percent
and the interest is compound is monthly
well let's put the numbers in
to our old buddy the formula
we know we're gonna put six hundred dollars in originally were to the 1200
come from well it said it doubled it so I double dip
so apparently that's what my accrued would be obviously if I double six
hundred dollars
I put twelve percent in for the interest rain
and 12 in for the up
an because it's compound in monthly
I'm sposed to find the key I wanna know how long to do this
on that team is gonna be a problem because he's up in the exponent
well let's start solving
I know I can get rid of the 600 by dividing both sides
okay and 1200 divided by six hundred is to
and I think I can do what's in parentheses okay and
and one
and point zero one but I still have a problem here I've got 12t
up in the exponent kinda can reach him can I what would you do to both sides
well
and love you remember but when we have something of this form
here's where you're going to use the logs note that has
variable were looking for in the exponent so that's 1 I've got to get you
used to taking the log on both sides now what's the advantage of that
member the properties alongs
going to allow us to take that 12t down
in front where we can do something about it K
now who's keeping tea from being alone
or the 12 whose multiplying
and the log love 1.01 who's also multiplying so what I could do
is divided by them
and I'll get I can put that into my calculator the log off to
divided by the log 1.01
now I have to do this figure out what that number is
and then divide divide both sides by 12
and seventy divided by 12
is 5.83 know that the question asked how long
so the answer is five-point 83 years pretty cool couldn't have done it
without logarithms
now in banking other than the amount you've invested
the time is about the only thing you have any power over you have no power
over the interest rate
certainly the formula controls the accrued so
you know these are the kinda questions we'll be asking how much should I invest
or how much time should I invest it for
let's try another one how long will it take an investor
earn two hundred dollars %uh interest to passed in a different way here
turned two hundred dollars have interest with an investment
a fifteen hundred dollars that's a principal at an interest rate of 10
percent
when the interest rate is compound it's semi
annually that's twice a year well once again let's use this formula and put in
the numbers
I put in the 1500 for the principal
the 1700 came from adding
200 to the 1500 because I earned two hundred
I love interest with call that the accrued is principal
plus interest and then as you can see your blue 10 percent
and your to for the N
and what I don't know is time
so let's divide first to get rid of the 1500
and now reduce that fraction okay now I'm gonna leave it like this
I'll show you why in a second what I mean is I'm not gonna divide
15 in the seventeen just yet okay pay attention
now I'm gonna do within parentheses I can do that with my calculator one-plus
a .1 0 divided by two
is 1.05 now
since I'm looking for something in the exponent the
T that red t I can't get to it
unless I use the logs
okay remember what the log the property of logarithms allows me to do
it allows me to bring that to T down in front where I can do something about it
now got the log a 17-15 so on the left
and on the right I've gotta get t loan getting rid of who's keeping from me
it from being alone is the two the brown to and the log of 1.05 and both of them
are multiplying so I'm gonna divide
both sides by I do it in one step
and I'll get rid of the two and it'll get rid of the longer 1.05 and leave me
with T
on it now I've purposely not done
any calculations on the left in all that nightmare
because when I want you to do get used to
if you can use your calculator every time you hit enter on your calculator
it gives you an estimate so if you have to hit enter four times
you gonna be of well four times as much
we want to try and do our calculations
by only hitting Enter wants so let's try and do this
in one step on the calculator so not to truncate
or lose accuracy in the calculator and the answer you get
is 1.28 years it's gonna be ok.
a tough one to put into the calculator but would you need to know is
always put the whole numerator in parentheses
the whole denominator in parentheses any longer than it should have their own set
of parentheses
and finally any exponent should have
his own set of parentheses that's four things and
when you're done when you're about to hit an hour make sure you have an even
number apprentice ease
because every beginning privacy should have an ending pregnancies
thats a numerator should have its own parentheses
a denominator should have its own parentheses any longer than it should
have
it's set of parentheses and any
exponent should have it set up parentheses that's going to be the tough
part in this case
okay give it a go remember any time you're solving for something
in an exponent you're gonna be somewhere taking a longer both sides