In this article we will prove the logarithm rule:
\[\begin{align*} \boxed{\log_{b}(b^k) = k} \end{align*}\]
Based on how the logarithms work, $b$ must be positive and not equal to $1$, and $k$ can be any real number.
The above rule is saying that if we take the logarithm of a number that is to the $k^{th}$ power of the base $b$, the result is simply the exponent $k$. In other words:
\[\begin{align*} \log_{b}(b^k) = k \end{align*}\]
This rule works because the logarithm $\log_b(x)$ asks, “What power must we raise $b$ to in order to get $x$?”. As an equation, this looks like this:
\[\begin{align*} \log_{b}(b^k) = x, \end{align*}\]
Due to the definition of a logarithm, we can rearrange the equation to look like this by raising both sides of the equation to the $b^{th}$ power:
\[\begin{align*} b^k = b^x. \end{align*}\]
Then, this means that,
\[\begin{align*} k = x. \end{align*}\]
Substituting back into our original equation we find that:
\[\begin{align*} \log_{b}(b^k) = k. \end{align*}\]
This proves that the below logarithm rule is true:
\[\begin{align*} \boxed{\log_{b}(b^k) = k} \end{align*}.\]
Throughout Precalculus and Calculus, logarithms will appear frequently, and this is one of the most important rules for solving those types of problems.
