Suppose we want to expand the following log as much as possible:
\begin{equation*}
\log_7\left(\dfrac{x^5\sqrt{y}}{z^4}\right)
\end{equation*}
It may be tempting to bring some of the exponents out, but we can only do that when the exponent is on the entire inside. So, for example, we cannot yet bring down the 5 because right now itβs only the exponent on part of the inside (just the \(x\)). But, if we separate this into multiple logs, then we will have each exponent on its own piece. So, letβs start by using the first two rules to separate the logs.
\begin{equation*}
\log_7\left(x^5\right)+\log_7\left(\sqrt{y}\right)-\log_7\left(z^4\right)
\end{equation*}
Now, we will want to bring our exponents outside. However, the middle term has a square root instead of an exponent. So, we need to convert \(\sqrt{y}\) into \(y\) to a power. Remember that the square root is the same thing as the \(1/2\) power
\begin{equation*}
\log_7\left(x^5\right)+\log_7\left(y^{1/2}\right)-\log_7\left(z^4\right)
\end{equation*}
Now we are finally ready to get our answer by bringing down the exponents.
\begin{equation*}
5\log_7\left(x\right)+\frac{1}{2}\log_7\left(y\right)-4\log_7\left(z\right)
\end{equation*}