The logarithm is the answer to the question "to what power do I need to raise a number to get the result?" For example, the equation $2^x = 8$ can be solved as $x = \log_28 = 3$. Use [[logarithm rules]] to solve logarithm equations. ## applications of the logarithm Use the logarithm when solving for an exponent. For example, the population at time $t$ given a growth rate $k$ can be modeled as $P(t) = P_0e^{kt}$. Let's say we know that after 5 years the population doubled. What is the population growth rate? We can take the natural log (log with base $e$) to simplify this equation. $\displaylines{2P_0 = P_0e^{5k} \\ 2 = e^{5k} \\ \ln2 = 5k \\ \frac{\ln2}{5}=k}$