1. We are given the function $h(x) = \ln(g(x))^3$ and need to find $h'(2)$ given $g(2) = 5$ and $g'(2) = -3$. 2. First, rewrite the function using logarithm properties: $h(x) = 3 \ln(g(x))$ because $\ln(a^3) = 3 \ln(a)$. 3. To find $h'(x)$, use the chain rule. The derivative of $3 \ln(g(x))$ is: $$h'(x) = 3 \cdot \frac{1}{g(x)} \cdot g'(x)$$ 4. Substitute $x=2$, $g(2) = 5$, and $g'(2) = -3$: $$h'(2) = 3 \cdot \frac{1}{5} \cdot (-3) = \frac{3 \times (-3)}{5} = -\frac{9}{5}$$ 5. Therefore, the value of $h'(2)$ is $-\frac{9}{5}$. The correct answer is option c.