Chapter 8.5, Problem 65HP

To determine

To find: The expression that does not belong to the given expressions and explain.

Answer to Problem 65HP

The expression ${\log }_{b}24={\log }_{b}20+{\log }_{b}4$ is not true.

Explanation of Solution

Given:

The given expressions are as follows.

  $\begin{array}{c}{\log }_{b}24={\log }_{b}2+{\log }_{b}12\\ {\log }_{b}24={\log }_{b}20+{\log }_{b}4\\ {\log }_{b}24={\log }_{b}8+{\log }_{b}3\\ {\log }_{b}24={\log }_{b}4+{\log }_{b}6\end{array}$

Calculation:

Solve the expression ${\log }_{b}24={\log }_{b}2+{\log }_{b}12$ by using Product property.

  $\begin{array}{c}{\log }_{b}24={\log }_b\left(2\cdot 12\right)\\ ={\log }_{b}2+{\log }_{b}12\end{array}$

Solve the expression ${\log }_{b}24={\log }_{b}20+{\log }_{b}4$ by using Product property.

  ${\log }_{b}24\ne {\log }_{b}20+{\log }_{b}4$

Solve the expression

  ${\log }_{b}24={\log }_{b}8+{\log }_{b}3$ by using Product property.

  $\begin{array}{c}{\log }_{b}24={\log }_b\left(8\cdot 3\right)\\ ={\log }_{b}8+{\log }_{b}3\end{array}$

Solve the expression ${\log }_{b}24={\log }_{b}4+{\log }_{b}6$ by using Product property.

  $\begin{array}{c}{\log }_{b}24={\log }_b\left(4\cdot 6\right)\\ ={\log }_{b}4+{\log }_{b}6\end{array}$

Therefore, the expression ${\log }_{b}24={\log }_{b}20+{\log }_{b}4$ is not true.