Chapter 7.3, Problem 33PPS

To determine

To Evaluate: The expression.

Answer to Problem 33PPS

  ${\log }_{\frac{1}{5}}3125=-5$

Explanation of Solution

Given: ${\log }_{\frac{1}{5}}3125$

Formula used:

  $\begin{array}{l}{\log }_{b}x=yifandonlyif\\ b^y=x\end{array}$

Given: ${\log }_{\frac{1}{5}}3125$

Let the expression is equals to $y$

Then

Using the logarithm form

  ${\log }_{b}x=yif{b}^y=x$

  $\begin{array}{l}{\log }_{\frac{1}{5}}3125=y\\ b=\frac{1}{5},x=3125\\ then\\ {\frac{1}{5}}^y=3125\\ 5^{-1y}=5^5\\ -y=5\\ y=-5\\ {\log }_{\frac{1}{5}}3125=-5\end{array}$

Hence, ${\log }_{\frac{1}{5}}3125=-5$