Chapter 11.4, Problem 10CFU

To determine

To evaluate: The expression ${\log }_2\frac{1}{16}$ .

Answer to Problem 10CFU

The value of ${\log }_2\frac{1}{16}$ is -4.

Explanation of Solution

Consider the given expression.

  $\begin{array}{c}{\log }_2\frac{1}{16}={\log }_2{\left(16\right)}^{-1}\\ ={\log }_2{\left(2^4\right)}^{-1}\\ ={\log }_2{\left(2\right)}^{-4}\end{array}$

By the Power Property of logarithms, ${\log }_2{\left(2\right)}^{-4}=-4{\log }_{2}2$ . So,

  $\begin{array}{c}{\log }_2\frac{1}{16}=-4{\log }_{2}2\\ =-4\times 1\left[\because {\log }_{a}a=1\right]\\ =-4\end{array}$