Question 242151
After tripling $500 we will have $1500. So the equation we have to solve is:
{{{1500 = 500e^(0.0825t)}}}<br>
First we'll divide both sides by 500:
{{{3 = e^(0.0825t)}}}
Now, since the variable is up in the exponent, we will use logarithms to solve for it. The natural choice (excuse the pun) is to use natural logarithms (aka ln) because e is the base:
{{{ln(3) = ln(e^(0.0825t))}}}
Using the property of logarithms, {{{log(a, (p^q)) = q*log(a, (p))}}}, we can move the exponent in the argument out in front as a coefficient:
{{{ln(3) = (0.0825t)ln(e)}}}
Since {{{ln(e) = 1}}} by definition we now have:
{{{ln(3) = (0.0825t)}}}
Divide both sides by 0.0825:
{{{ln(3)/0.0825 = t}}}
This is an exact answer. You can use your calculator to find a decimal approximation for t (which is the time it will take to triple your money).<br>
If your calculator doesn't "do" ln, we can use base 10 logs instead of ln:
{{{log((3)) = log((e^(0.0825t)))}}}
{{{log((3)) = (0.0825t)log((e))}}}
Since we used base 10 logs, log(e) will not "disappear" like ln(e). So we are "stuck" with it. Divide both sides by 0.0825log(e):
{{{log((3))/(0.0825*log((e))) = t}}} (e is approximately equal to 2.7182818284590451. Round this off as you choose and then use your calculator to find t.<br>
Although {{{log((3))/(0.0825*log((e)))}}} and {{{ln(3)/0.0825}}} do not look the same they are actually equal.