Q 44. - Precalculus Enhanced with Graphing Utilities

Question

Find the domain of the function.

$g\left(x\right)=\ln \left(\frac{1}{x-5}\right)$

Step-by-Step Solution

Answer

The domain of the given function $g\left(x\right)$  is $\left(5,\infty \right)$.

Step 1. Given information

A logarithmic function is given as $g\left(x\right)=\ln \left(\frac{1}{x-5}\right)$.

Step 2. Formula Used

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

Step 3. Finding The Domain

The function is $f\left(x\right)=\ln \left(\frac{1}{x-5}\right)$

Now, for the domain,

$\frac{1}{x-5}>0$

On solving

$$\begin{gathered} \frac{x-5}{{\left(x-5\right)}^2}>0 \\ x-5>0 \\ x>5 \end{gathered}$$

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

interval notation, is $\left(-5,\infty \right)$.