Question

$$\begin{gathered} f\left(x\right)={\log }_4(x-2) \\ \left(a\right)\mathrm{Solve}f\left(x\right)=2 \\ \left(b\right)\mathrm{Solve}f\left(x\right)\le 2 \end{gathered}$$

Short answer

Part(a) The solution set is {18}

Part(b) The solution set is$(2,18]$

Step-by-step solution

Part(a) Step 1. Given Information

The given function is$f\left(x\right)={\log }_4(x-2)$

Part(a) Step 2. Calculation

Substitute 2 in the given function,

$$\begin{gathered} 2={\log }_4(x-2) \\ x-2=4^2 \\ x-2=16 \\ x=18 \end{gathered}$$

Part(b) Step 1. Calculation

The domain of function consists of all x for which $x-2>0orx>2$

Replace the function with ${\log }_4(x-2)$in the given inequality.

$$\begin{gathered} {\log }_4(x-2)\le 2 \\ x-2\le 4^2 \\ x-2\le 16 \\ x\le 18 \end{gathered}$$

Thus,$x>2andx\le 18$