Chapter 10, Problem 5CR

(a)

To determine

To calculate: The solution of the equation $f(x)=2$, where $f(x)={\log }_4(x-2)$

Answer to Problem 5CR

Solution:

The solution set for the equation $f(x)=2$ is $x=18$

Explanation of Solution

Given information:

The function $f(x)={\log }_4(x-2)$

Formula used:

Logarithmic property: If ${\log }_{a}b=c$, then $b=a^c$

Calculation:

Consider the equation $f(x)=2$

$\Rightarrow {\log }_4(x-2)=2$

Using logarithmic property,

$\Rightarrow 4^2=\left(x-2\right)$

$\Rightarrow 16=x-2$

$\Rightarrow x=18$

Thus, the solution set for the equation $f(x)=2$ is $x=18$

(b)

To determine

To calculate: The solution of the inequality $f(x)\le 2$, where $f(x)={\log }_4(x-2)$

Answer to Problem 5CR

Solution:

The inequality $f(x)\le 2$ has the solution $\left(2,18\right]$

Explanation of Solution

Given information:

The function $f(x)={\log }_4(x-2)$

Formula used:

Logarithmic property: If ${\log }_{a}b\le c$, then $b\le a^c$

Calculation:

Consider the equation $f(x)\le 2$

$\Rightarrow {\log }_4(x-2)\le 2$

Using logarithmic properties,

$\Rightarrow (x-2)\le 4^2$

$\Rightarrow x-2\le 16$

$\Rightarrow x\le 18$

Logarithm is defined on positive real axis.

Then in $f(x)={\log }_4(x-2)$, $\left(x-2\right)>0$

$\Rightarrow x>2$

Combining the inequalities $x\le 18$, $x>2$, the solution of the inequality $f(x)\le 2$ is $\left(2,18\right]$.