Question

I have an initial investment of $1000 at an interest rate of 5%, compounded continuously. How long will it take my investment to double? Using the natural logarithmic formula, how do I solve for t (time)?

Answer 1

You need the continuously compounding formula first.
\[A = P e ^{rt}\]
where P = principal amount (initial investment)
r = annual interest rate (as a decimal)
t = number of years
A = amount after time t

Answer 2

How do I solve for t?

Answer 3

Since you want your money to double you want to end up with $2000 right?
So your A = 2000, P = 1000, r = 0.05.
Because t is in the exponent, you will need to natural log both sides and use the power rule to solve for the "t" .

Answer 4

Can you show me how that looks?

Answer 5

How about you have a guess.

Answer 6

Plug in the values of what you know so far if you're not sure.

Answer 7

Thanks for your help.

Answer 8

What did you get so far? Did you swap the A for 2000, the P for 1000 and the r for 0.05?

Answer 9

Yes I did and that's as far as I got.

Answer 10

OK, now let's get this t alone. Divide by 1000.

Answer 11

Now I have 2=e^.05t

Answer 12

Excellent! No if you literally write "Ln" in front of both sides of the equation, that is, in front of the 2 and in front of the e^0.05t, then the "Ln" and the "e" cancel each other out and the 0.05t drop down to the numerator!

Answer 13

Therefore t=40?

Answer 14

Did you do ln(2)/0.05? I think you may have left out the natural log on the 2.

Answer 15

No I did not. That gives me 13.86

Answer 16

When I put 13.86 back into my original equation, it checks out. Thanks for your help!

Answer 17

You're very welcome!