= Solution
Write $\tau=T-t$ and $a=e^{-\alpha\tau}$. The <stochastic integral> in <3/b> has a deterministic integrand, so it is a <Gaussian random variable> independent of the information available at $t$. Its <variance>, by the <Itô isometry>, is
$$
v=\sigma^2 e^{-2\alpha T}\int_t^T e^{2\alpha u}du=\frac{\sigma^2}{2\alpha}(1-a^2).
$$
Thus $x_T$ conditionally has a <normal distribution> with <expected value> $m=ax_t+(1-a)\widetilde\mu$ and <variance> $v$. The <moment-generating function of a normal distribution> gives $\mathbb E_t[e^{x_T}]=e^{m+v/2}$, so the <conditional mean of a log-mean-reverting price> is
$$
\boxed{F(t,T)=\exp\!\left[e^{-\alpha(T-t)}\log S_t+\widetilde\mu(1-e^{-\alpha(T-t)})+\frac{\sigma^2}{4\alpha}(1-e^{-2\alpha(T-t)})\right]}.
$$
The correction $v/2$ is essential: exponentiating the conditional mean of the logarithm alone would omit the dispersion of the <lognormal distribution>.
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