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So if we invest $2000 at 6% compounded monthly, how long will it take the
account to double in value? So, we know how to set up sort of our
basic compound interest equation. We know that the amount we have after n
years will be Let's my initial amount is 2,000, my rate is 6% compounded monthly
for I don't know how many years. So I know my initial amount is 2,000, r
over k to the k times Tn and I don't know how many years it is.
So this is a little different because I don't know how many years it is.
So I'm not trying to figure out how much I'll have in some number of years instead
I know how much I want, and are trying to figure out how long.
So the how long is saying, I want n, I want to know what n is.
Now, in order for my account to double in value, it's going to have to end up being
worth 4,000, twice as much. And so to answer this question, I'm
going to set the amount that I end up with equal to 4,000 oops, I've got one
too few zeroes there, there we are and, and now I can start solving for n.
Now, the first thing I need to do is get that exponential by itself, so we're
going to divide both sides of the equation here by 2000 and then the left
site, 4,000 over 2,000 is just 2 equals 1.005 to the 12n.
So now, in order to solve for our exponent we're going to need to use the
logarithm. So we can take the log of both sides,
apply the log function to both sides and now we can use the log property, which
says that the exponent gets pulled down in front.
And we're going to go ahead and pull down the entire exponent here just for
simplicity. So we got log of 2 equals 12 times n
times log of 1.005. Now at this point there's a couple routes
we can go. I think I'm just going to go let's pull
out our calculator route. So I'm going to start on the left with
log of 2 is 0.301. So this is 12 times n times, let's see
that's 1.005 log is .00217 remember to keep 3 significant digits here so .00217,
.00217. Now just for simplicity, I am going to go
ahead and multiply together my 12 and my 0.00217.
and since I already have this in my calculator I'm just going to say times 12
gives me 0.0260, 0.0260. Okay 0.0260 times n.
So that was the 1 times the 0.0217, and now I can divide by, divide by that.
So I've got 0.301 divided by 0.0260 is 11.577, okay.
So n is 11.5, what did I just say, 11.577, 77 years.
So it's going to take a little over eleven and a half years for this account
to double in value.