## College Algebra (10th Edition)

Domain: $x\gt -1.5$ or $(-1.5, +\infty)$.
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 $(-1.5, +\infty)$.