\[\begin{align*} \boxed{\log_{b}(b^k) = k} \end{align*}\]
This rule says that if we take the log 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 $x$?” So when we write:
\[\begin{align*} \log_{b}(b^k) = x, \end{align*}\]
due to the definition of a logarithm, we can rearrange the equation to look like this:
\[\begin{align*} b^x = b^k. \end{align*}\]
Then, this means that,
\[\begin{align*} x = k. \end{align*}\]
Substituting back into our original equation we find that:
\[\begin{align*} \log_{b}(b^k) = k. \end{align*}\]
This proves that:
\[\begin{align*} \boxed{\log_{b}(b^k) = k} \end{align*}\]
