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I am also doing the same exercise. Let $GL(t,s)$ denote the value of the Golden Logarithm at time $t$ when the underlying stock has price $s$. As in the answer by bcf, the Black-Scholes model gives: $$GL(t,s) = e^{-r(T-t)}\mathbb{E}^Q[GL(T)]=e^{-r(T-t)}\mathbb{E}^Q[\log S(T)],$$ with expectation taken over the risk-free measure. The stochastic process ...
If $X$ is a lognormally distributed variable, $X = e^{\mu + \nu Z}$, so $\ln X$ has mean $\mu$ and variance $\nu^2$, and $Z$ is normally distributed, then $$E\left[ X^n \right] = e^{n\mu + \frac{1}{2} n^2\nu^2}$$ This solves your question with $X = S_T/S_t$, $n = \beta$, $\mu = (r -\frac{1}{2} \sigma^2) (T-t)$ and $\nu = \sigma \sqrt{T-t}$ in the Black ...