If you invest $100 each month into an 
account earning 3% compounded monthly,
how long will it take the account to grow 
to $10,000?
you might notice that this initial 
scenario is describing an annuity
problem. 
The annuity would be set up with a
starting amount of $100. 
We're growing at 3% compounded monthly
for, I don't really know how many years 
yet?
right, so we've got a initial value of 
100, we've got our rate of 3% and our
compounding monthly, and we don't know 
how many years we're talking about.
Okay, but, so now in order to answer this 
question what we're really wondering is
when will the amount we end up with be 
10,000.
So we are going to stick 10,000 on the 
left here.
Now, 0.03 over 12 is 0.0025 to the 12n 
minus 1 over 0.0025.
Okay, so now, let's start solving for n. 
Now, this is a little bit tricky of a
process here because there's a lot of 
things going on.
But let's see if we can work our way 
through it.
So the first thing we're going to do is 
get rid of this fraction here by
multiplying both sides by 0.0025. 
So we're going to multiply this side by
0.0025 as well. 
On the right hand side, this is going to
reduce or cancel with the denominater 
there.
On the left hand side, we'll go ahead and 
multiply 10,000 by 0.0025 is, is 25.
So we've got 25 equals 100 times 1.0025 
to the 12n minus 1.
So now let's get rid of that 100, so 
we're going to divide both sides by 100,
which again will reduce it on the right. 
On the left hand side, 25 divided by 100
is 0.25 So we've got 0.25 equals 1.0025 
to the 12n minus 1.
Now, again we're trying to get that 
exponential by itself here so let's go
ahead and add 1 to both sides. 
So now we've got 1.25 equals 1.0025 to
the 12n. 
I need to get some of this out of the
way, so hold on. 
Let me get some of this out of the way.
Now, we have the exponential by itself 
there so now we can apply the logarithm
to both sides of the equation. 
And the logarithm lets us take that
exponent and pull it down in front. 
So we'll have log of 1.25 equals 12 and
times the log of 1.0025. 
So now we can pull out our calculators.
So, on here we're got 1.25 logorithm is 
0.06 969, so 0.0969.
On the right hand side ,we've got 1.0025 
log of that.
Now, I'm going to go ahead and multiply 
that by 12 now, so times 12 is 0.0130.
0.0130, okay, 0.0130 at n. 
So now I can divide both sides by 0.013,
0.013. 
and, see what I end up.
So we got 969, divided by 0.013 is 7.454. 
So no is 7.454.
And I say 454, 454 years. 
So, it's going to take a little over or a
little under seven and a half years for 
this account to grow to $10,000.
