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 logarithm; solving for a variable that's up in the exponent position
when you really can't reach it.
Now let me give you a scenario where we're going to use it
Consider the general interest formula for multiple compounding. Now this involves
your money at the bank.
"A" stands for the accrued amount that you get; the total amount of principal AND
interest over the years if you've invested "P" amount of principle.
That's the amount that you had put in originally. "r" is the (annual) interest rate.
"t" is the amount of time that you have the money invested (in years) and "n" is the number of periods
per year. For instance "quarterly" "n" would be four, or "monthly" "n" would be 12, etc
Now before you get nervous let's answer a question using this formula without
using logarithms. Consider asking ...how much would you (the investor) accrue, how much money
would be in your account if you invested five hundred dollars at an interest rate
of seven percent for four years (you kept it in the bank for four years) when the
interest is (as it typically is) , is compounded monthly? Well let's put the
numbers in. Our principal is 500 as you can see. .07 is seven percent (for the
interest) and the magic number for monthly is 12 and the time, t is 4.
All we have to do is use this formula and basically I would just put
this into a calculator.
make no mistake, because there's no solving to do. And this is what you would have to type.
Please note that you have to put the exponent; "12 times 4" in its own set
of parentheses. Anytime you have an exponent it has to be in parentheses.
Be careful with parentheses or you'll get an answer but it just won't be right.
And the answer, in this case, is quite believeable... After four years, 500
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.
Ok let's start off again... How much would an investor accrue...
how much would they have in their bank account after throwing down 3000
dollars at an interest rate of ten percent. That's a hefty interest rate
... for five years? So it's in the bank for five years, and in this case the interest is
compounded quarterly? So that means the magic number is 4. So we'll put in all
our numbers. 3000 four principal "P". Ten percent for interest rate "r"...
4 for the "n", and (of course) 5 for the time "t" in years. And if we
put this into our our calculator it should basically look like this.
Once again I warn you to make sure that you put the exponent in parentheses. Ok?
And we earned a lot of free money there. Remember you started off with 3,000
You had a high interest rate (10%) and you waited five years. But if you think about it...
Your 3000 dollars turned into 4900 something so basically you got about 1900
dollars of free money (that's what I call it anyway).
Ok, so that's how this formula works. But... let's get back to... What are we going to use the
logarithms for? Let's look at the same formula but we're going
to 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 of 600 dollars
if you had an interest rate of twelve percent and the interest is
compounded monthly? Well let's put the numbers into our old buddy the formula.
we know we're going to put 600 dollars in originally but where did the 1200
come from? Well it said we doubled it (the 600) so I doubled it.
So apparently that's what my accrued would be (if I doubled 600).
I put twelve percent in for the interest rate and 12 in for
the "n" because it's compounded monthly. I'm supposed to find the "t". I want to know...
how long it will take to do this. Well that "t" is going to be a problem because he's up in the
exponent. Well let's start solving.
I know that I can get rid of the 600 by dividing both sides.
Ok, and 1200 / 600 equals 2, And I think I can do what's in parentheses and add
1.01 but I still have a problem here. I've got "12t" up in the exponent.
I can't reach him can I? What would you do to both sides?
Well I hope you remember that when we have something of this form...
here's where you're going to use logarithms. Note that it has the variable that we're
looking for in the exponent. That's when I've got to get you used to...
taking the logarithm of both sides. Now what's the advantage of doing that?
Do you remember the 3rd property of logs?
It's going to allow us to take that "12t" down in front where we can do something
about it.
So now we ask...who's keeping "t" from being alone?
Well the 12, whose multiplying, and the log(1.01) , who's also multiplying...
so what I could do...
is divide both sides by them.
and I'll get... I can put that into my calculator... log(2) divided by the log(1.01)
And now all i have to do is figure out what that number is and then
divide divide both sides by 12. And 70 divided by 12 is... 5.83 Now the question asked...
how long?
So the answer is "5.83 years". Pretty cool! And we 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 these are the kind
of questions we will be asking: How much should I invest? or... How much time should
I invest it for? How long will it take?
Ok let's try another one. How long would it take an investor to earn $200
dollars (of interest) with
with an investment of $1,500 (that's the principal) at an interest rate
of ten percent when the interest rate is compounded semi-annually (that means 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
200 of interest. We call that the sum "accrued" (principal plus interest) and
then, as you can see, your blue ten percent and your 2 ( for the "n") and what I
don't know is time "t".
So let's divide first to get rid of the 1500.
And then I'll reduce that fraction.
Ok now I'm going to leave it like this.
I'll show you why in a second. What I mean is I'm NOT going to divide 15 into 17
just yet. Okay... Pay attention. Now I'm gonna do what's inside the parentheses. I can do
that with my calculator. 1 + .10 over 2 equals 1.05 Now, since I'm looking for
something in the exponent the red "t"
I can't get to it unless I use logarithms.
Remember what the 3rd property of logarithms allows me to do? It allows me
to bring that "2t" down in front where I can do something about it. Now
I've got the log(17/15) on the left. And on the right I want to get "t" alone by
getting rid of who's keeping it from being it from being alone. And that would be the (brown) 2,
and the log(1.05) and both of them are multiplying.
So i'm going to divide both sides by... (I'll do it in one step)...
and that'll get rid of the 2 and it will also get rid of the log(1.05) and leave me
with "t" won't it? Now I've purposely not done any calculations on the left yet. And
that's because, what I want you to do is get used to... if you're going to...
use your calculator, You need to know that every time you hit ENTER on your calculator it gives you an
ESTIMATE. So if you have to hit ENTER 4 times you're going to get an inaccurate
answer. So we want to try to do our calculations by only hitting
ENTER once. So let's always try and do this in one step on the calculator, so not to
truncate or lose accuracy in the calculator. And the answer we get is...
1.28 years. Now it's going to be a a tough one to put into the calculator but what
you need to know is... ALWAYS put the whole numerator in parenthesis, the whole denominator
in parentheses, any logarithms should have their own set of parentheses,
and finally... any exponent should have its own set of parentheses.
That's 4 things and when you're done, when you're about to hit ENTER...
Make sure you have an even number of parentheses because every beginning
parentheses should have an ending parentheses. That's... a numerator should
have its own parentheses, a denominator should have its own parentheses. any
logarithm should have its own set of parentheses and any exponent should have
its own set of parentheses. That's going to be the tough part in this case.
Okay? Give it a go.
And remember... Anytime you're solving for something in an exponent you're going to be, at some point,
taking the log of both sides of the equation.