Chapter 3.4, Problem 94E

To determine

Determine the actual solution.

Answer to Problem 94E

The actual solution is $x=9$

Explanation of Solution

Given:

The logarithm of the product of two numbers is equal to the difference of the logarithms of the numbers.

Consider the equation ${\log }_{3}x+{\log }_3(x-8)=2.$

Write it as

  $\begin{array}{ccc}{\log }_{3}x+{\log }_3(x-8)& =2& \\ {\log }_3[x(x-8)]& =2& \text{By the product property}\text{.}\end{array}$

  $$

Solving it algebraically,

  $\begin{array}{cccc}{\log }_3[x(x-8)]& =2& & \\ 3& {\log }_3[x(x-8)]& =3^2& \text{Exponentiate each side with base}3\end{array}$

  $x(x-8)=9$

  $x=9$

  $x=-1$

But $x=-1$ is extraneous. So $x=9$ is the only solution which is clear from the graph given below.

  

Because the domain of a logarithmic function is all real numbers greater than 1 , the solution $x=-1$ is extraneous. So the actual solution is $x=9$