# Logarithm: Compound interest problem

## Homework Statement

How long does it take for a sum of money to double if it is invested at 8% compounded semi-annually?

## Homework Equations

A = P(1+i)n

A: Compounded amount
P: Initial amount
i: Interest rate
n: Period

## The Attempt at a Solution

A = P(1+i)n
(2x) = (x)[1+(0.08)]2n (2n because it's compounded semi annually)
2 = 1.082n (x cancels out)
2n = log1.082
n = (log1.082)/2
n = 4.5032 per half a year

Although the answer given at the back of the package is 8.836a.
And I'm assuming "a" stands for annual. So I'm not sure where I went wrong.
Any help is much appreciated! Thanks in advance!

## Answers and Replies

cepheid
Staff Emeritus
Gold Member
When you're compounding at some period other than annually, like say at intervals of 1/N years, the equation is:

A = P(1+i/N)n

and it's understood that n is the number of compounding periods, which means it represents the time interval in units of Nths of a year, rather than years.

Edit: In other words, you apply one Nth of the interest rate, and you do this N times a year.

In this case, we compound semi-annually, or every 1/2 year, so N = 2.

So, we have

log(A) = log(P) + nlog(1 + 0.08/2)

log(2P) = log(P) + nlog(1.04)

log(P) + log(2) - log(P) = nlog(1.04)

n = log(2)/log(1.04)

n = 17.6729876851

So, the total time required is 17.673 HALF-YEARS (compounding periods). Divide that by 2 to get 8.836 years. I believe the 'a' stands for 'annum', which is Latin for 'year'.

ohhhh okay thanks alot!