# How do you solve 223.11 = 1.04^t ?

Apr 21, 2016

Take the log of both sides, then solve for $t = 137.9$

#### Explanation:

Questions with variables in the exponent can usually be solved using the logarithm. When you use a log, you have a choice of base, i.e. the natural logarithm (base $e$), log base 10, etc. The choice doesn't affect the answer, but sometimes it is easier to choose a base that matches the base of the power in the question. In this case there doesn't seem to be an obvious choice, so let's use the natural log. Start by taking the log of both sides of the equation:

$\ln \left(223.11\right) = t \cdot \ln \left(1.04\right)$

The power in the expression becomes a multiplication factor after taking the log. Now we just need to solve for $t$

$t = \ln \frac{223.11}{\ln} \left(1.04\right) = \frac{5.048}{0.0392} = 137.9$