Question

Solve this application using logarithms. How long will money in savings take to double at 5% interest compounded annually? Round the answer to the nearest hundredth of a year.

Answer 1

This one uses the formula A(t)=P(1+r)^t, where A(t) is the amount in the end, P is the initial investment, r is the rate, and t is the time in years. Because we are compounding annually, the formula is simpler than it would be if we had to consider half of a year or a quarter of a year. If we don't know how much money we started with, how can we know what its double is you might ask. Let's make up the numbers!! We will start with 1000, so doubling it is 2000. So the formula would look like this: 2000=1000(1.05)^t. Divide thee 2000 by the 1000 to get 2 = 1.05^t. Now take the common log of both sides to get
log 2 = log 1.05^t. One of the properties of logs tells us that we can bring the t down in front of the log so it looks like this now: log 2 = t log 1.05.  Divide both sides by log 1.05 to get the t alone: log 2/log 1.05 = t.  Do the math on your calculator to get that t = 14.2 years.