The optimal leverage which maximizes the log utility of a portfolio is well known and has a simple solution.
For example, for a Geometric Brownian Motion with drift a and volatility b, the optimal leverage is $\frac{a}{b^2}$
Has there been any work on this when options are involved?
For example, assume we also have 1 at-the-money put option available to trade. What are the optimal weights for log utility and is Monte Carlo the only way to find this? There may not be an analytical solution but has anyone studied this?
What you allude to here is the optimal allocation to a single risky asset with expected return $a$ and variance $b^2$. Assuming a risk free rate of $r$ (which you appear to take as $0$), the allocation weight to the risky asset that maximizes constant relative risk aversion (CRRA) utility with parameter $\lambda$ is
$$w^* = \frac{a-r}{\lambda b^2}$$
This is, in fact, what you get from mean-variance optimization
$$\max_w r + w(a-r) - \frac{1}{2}\lambda w^2 b^2,$$
where $\lambda$ is a Lagrange multiplier.
The case where $\lambda = 1$ corresponds to maximization of a log utility function, as you say. In effect, the maximization achieves an optimal trade off between return and risk when risk is (narrowly) defined as portfolio volatility.
It is well known that consideration of only the mean and variance is not appropriate for portfolios that include options with their highly asymmetric return distributions.
The best reference to begin with on this topic is this paper by Hayne Leland:(https://www.researchgate.net/publication/279937457_Beyond_Mean-Variance_Risk_and_Performance_Measurement_in_a_Nonsymmetrical_World