Q 41. - Precalculus Enhanced with Graphing Utilities

Question

Find the domain of the function.

$f\left(x\right)=3-2{\log }_4\left(\frac{x}{2}-5\right)$

Step-by-Step Solution

Answer

The domain of the given function $f\left(x\right)$ is $\left(10,\infty \right)$.

Step 1. Given information

A logarithmic function is given as $f\left(x\right)=3-2{\log }_4\left(\frac{x}{2}-5\right)$

Step 2. Formula Used

The domain of the logarithmic function $f\left(x\right)={\log }_a\left(x\right)$ is determined by $x>0$. 

Step 3. Finding The Domain

The function is $f\left(x\right)=3-2{\log }_4\left(\frac{x}{2}-5\right)$

Now, for the domain,

$\frac{x}{2}-5>0$

On solving

$$\begin{gathered} x-10>0 \\ x>10 \end{gathered}$$

The required domain of $f\left(x\right)$, using

interval notation, is $\left(10,\infty \right)$.