Chapter 6.6, Problem 46E

To determine

To argue whether a logarithmic equation have a negative solution or not.

Answer to Problem 46E

Friend is not correct since a logarithmic equation can have a negative solution.

Explanation of Solution

Given information:

It is argued that a logarithmic equation cannot have a negative solution because logarithmic functions are not defined for negative numbers.

The statement is false since a logarithmic equation can have a negative solution as described below:

  $\begin{array}{c}{\log }_4\left(x^2+12x\right)={\log }_4\left(64\right)\\ x^2+12x-64=0\\ x^2+16x-4x-64=0\\ \left(x+16\right)\left(x-4\right)=0\end{array}$

Thus, $x=-16$ or $x=4$ .

Hence, $x=-16$ is the negative solution of the logarithmic equation.