Answer
a) $\{18\}$
b) $(2,18]$
Work Step by Step
We are given the function:
$f(x)=\log_4 (x-2)$
a) Solve the equation:
$f(x)=2$
$\log_4 (x-2)=2$
$x-2=4^2$
$x-2=16$
$x=18$
Check the solution:
$\log_4 (18-2)\stackrel{?}{=} 2$
$\log_4 16\stackrel{?}{=} 2$
$2=2\checkmark$
The solution set is:
$\{18\}$
b) Solve the inequality:
$f(x)\leq 2$
$\log_4 (x-2)\leq 2$
$\log_4 (x-2)\leq \log_4 4^2$
$x-2\leq 16$
$x\leq 18$
As the function $f(x)=\log_4 (x-2)$ is defined for $x>2$, the solution set is:
$(2,18]$
