Answer
Domain:
$x\gt -1.5$ or $(-1.5, +\infty)$.
Work Step by Step
RECALL:
The logarithmic function $f(x) = a+b \cdot \ln{x}$ is defined only when $x \gt 0$. Thus, its domain is $x \gt 0$.
This means that the function $f(x) = 8+5\ln{(2x+3)}$ is defined only when $2x+3 \gt 0$.
Solve the inequality to obtain:
$2x+3 \gt 0
\\2x+3-3 \gt 0-3
\\2x \gt -3$
Divide 2 on both sides of the equation to obtain:
$\dfrac{2x}{2} \gt \dfrac{-3}{2}
\\x \gt -1.5$
Thus, the domain of the given function is $x\gt -1.5$.
In interval notation, the domain is $(-1.5, +\infty)$.
