Thanks to @Phun and @oliversm I solved the problem. So I'm posting here the solution in case someone will need it.
Under Black-Scholes assets dynamics are determined by a Geometric Brownian Motion, and we can define the price of a security at time $t+\Delta t$ as:
$$S_{t+\Delta t}=S_{t}\exp\left(\left(r-\frac{1}{2}\sigma^{2}\right)\Delta t+\sigma\sqrt{\Delta t}\varepsilon\right)\qquad\varepsilon\sim N(0,1)$$
defining $T=t+\Delta t$ and substituting above leads to:
$$S_{T}=S_{t}\exp\left(\left(r-\frac{1}{2}\sigma^{2}\right)\left(T-t\right)+\sigma\sqrt{T-t}\varepsilon\right)\qquad\varepsilon\sim N(0,1)$$
Now, under risk neutral probability pricing the drift term $\mu$ can be replaced with the interest rate, and setting $r=0$ leads to:
$$S_{T}=S_{t}\exp\left(-\frac{1}{2}\sigma^{2}\left(T-t\right)+\sigma\sqrt{T-t}\varepsilon\right)\qquad\varepsilon\sim N(0,1)$$
Following the procedure illustrated here, it is easy to show that:
$$\frac{S_{T}}{S_{t}}\sim\ln N\left(-\frac{1}{2}\sigma^{2}\left(T-t\right),\sigma^{2}\left(T-t\right)\right)
$$
or equivalently:
$$\ln S_{T}\sim N\left(-\frac{1}{2}\sigma^{2}\left(T-t\right),\sigma^{2}\left(T-t\right)\right)$$
At this point let's define $S = \ln (S_T / S_t) = \ln(S_T) - \ln(S_t)$. $S_t$ is known at time $t$, so we can add $\ln S_t$ to $S$. $S+\ln S_t$ will be normally distributed with mean:
$$\ln S_t-\frac{1}{2}\sigma^{2}\left(T-t\right)$$
and variance:
$$\sigma^{2}\left(T-t\right)$$
So:
$$S+\ln S_{t}\sim N\left(\ln S_{t}-\frac{1}{2}\sigma^{2}\left(T-t\right),\sigma^{2}\left(T-t\right)\right)$$
But since $S+\ln(S_t)=\ln(S_T)$ it follows that:
$$\ln S_{T}\sim N\left(\ln S_{t}-\frac{1}{2}\sigma^{2}\left(T-t\right),\sigma^{2}\left(T-t\right)\right)$$