Chapter 6.3, Problem 41E

To determine

Check the error and rewrite the expression in correct way.

Answer to Problem 41E

The correct expression is ${\log }_4\left(\frac{1}{64}\right)=-3$ .

Explanation of Solution

Given information:

The given expression is ${\log }_4\left(-3\right)=\frac{1}{64}$ .

Formula Used:

Logarithmic power rule is

If $a^b$ is an expression then logarithm form ${\log }_a{\left(a\right)}^b$ is equal to $b$ .

Consider the logarithm expression is ${\log }_4\left(-3\right)=\frac{1}{64}$ .

It is known that for a real base logarithm of a negative number is undefined.

So for the given expression, the term ${\log }_4\left(-3\right)$ is undefined.

Thus, the term ${\log }_4\left(-3\right)$ cannot be equal to $\frac{1}{64}$ .

Rewrite the expression in the correct way.

It is given that the exponential form is $4^{-3}=\frac{1}{64}$ .

Convert into logarithmic form. Use the logarithmic power rule.

  $\begin{array}{c}{\log }_4\left(\frac{1}{64}\right)={\log }_4\left(4^{-3}\right)\\ =-3\times {\log }_{4}4\\ =-3\times 1\\ =-3\end{array}$

Hence, the correct expression is ${\log }_4\left(\frac{1}{64}\right)=-3$ .