Question

$M=\frac{{\log }_{5}7}{{\log }_{5}8}$

Short answer

The obtained value is$\begin{array}{rcl}& & {\log }_{8}7\end{array}$.

Step-by-step solution

Step 1. Given information.

Consider the given information.

$M=\frac{{\log }_{5}7}{{\log }_{5}8}$

Step 2. Calculation.

Apply the base change property.

$\begin{array}{rcl}M& =& \frac{{\log }_{5}7}{{\log }_{5}8}\\ & =& \frac{\frac{\log 7}{\log 5}}{\frac{\log 8}{\log 5}}\\ & =& \frac{\log 7}{\log 5}·\frac{\log 5}{\log 8}\\ & =& \frac{\log 7}{\log 8}\\ & =& \frac{\log 7}{\log 8}\\ & =& {\log }_{8}7\end{array}$