# Math Help - word problem using logarithms

1. ## word problem using logarithms

can someone plese help me with this question, how to set it up and solve it. thanks!

a savings bond offers intrest at a rate of 6.6%, compounded semi-annually. suppose that a \$500 bond is purchased. how long will it take for the investment to double in value.?

2. If you're familiar with the basic 'single investment, compounding forward' set-up...

$500\left(1 \ + \ \frac{0.066}{2}\right)^n \ = \ F$

...where F is the amount the initial investment will grow to after n periods, the solution to your question starts with that basic layout:

$500\left(1 \ + \ \frac{0.066}{2}\right)^n \ = \ 1,000$

which immediately reduces to

$\left(1 \ + \ \frac{0.066}{2}\right)^n \ = \ 2$

Now solve for n. Start by taking the logs of both sides, and remember that $\text{ln}(a^b) \ = \ b\cdot \text{ln}(a)$. Finally, remember that in this set-up, n represents the number of semi-annual periods it'll take for the investment to double...so you might wanna re-express it as years in your final answer.

3. Please do not forget the remarkably reliable "Rule of 72s". For reasonable interest rates, the precision is almost always more than expected.

Since 6.6% is used semi-annually, we'll use 3.3% and calculate semi-annual periods.

72 / 3.3 = 21.818, just a little too high.