I've been reading about the Merton Portfolio problem and how you can use the HJB equation to solve it where
$0 = (∂_t + rx∂_x) h(t, x) − \frac{λ^2}{2} \frac{(∂_xh(t, x))^2}{∂_{xx}h(t, x)}$
with terminal condition $h(T, x) = U(x)$ and where $λ = \frac{µ−r}{\sigma}$
I've seen that the ansatz, $h(t, x) = −α(t) e^{−γ x β(t)}$ can be used to solve for the value function with exponential utility, but what about logarithmic utility? Would $h(t, x) = f(t)g(x)$ be the correct ansatz to solve for the value function with logarithmic utility?