Q 42. - Precalculus Enhanced with Graphing Utilities

Question

Find the domain of the function.

$g\left(x\right)=8+5\ln \left(2x+3\right)$

Step-by-Step Solution

Answer

The domain of the given function $g\left(x\right)$ is $\left(-\frac{3}{2},\infty \right)$.

Step 1. Given information

A logarithmic function is given as $g\left(x\right)=8+5\ln \left(2x+3\right)$

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 $g\left(x\right)=8+5\ln \left(2x+3\right)$

Now, for the domain,

$2x+3>0$

On solving

$$\begin{gathered} 2x>-3 \\ x>-\frac{3}{2} \end{gathered}$$

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

interval notation, is $\left(-\frac{3}{2},\infty \right)$.