Let $Y = \log X$, then:

$$\begin{align} Y &= Y0 + (\mu-\frac{\sigma^2}{2})t + \sigma Wt \\ EYt &=Y0 + (\mu-\frac{\sigma^2}{2})t \\ EYtEYs &= Y0^2 + Y0 (\mu-\frac{\sigma^2}{2}) (t+s) + (\mu-\frac{\sigma^2}{2})^2 t s \\ E(YtYs) &= E\left((Y0 + (\mu-\frac{\sigma^2}{2})t + \sigma Wt) (Y0 + (\mu-\frac{\sigma^2}{2})s + \sigma Ws)\right) \\ &= Y0^2 + Y0 (\mu-\frac{\sigma^2}{2}) (t+s) + (\mu-\frac{\sigma^2}{2})^2 t s + \dots + \sigma ^2 \min(t,s) \end{align}$$

What remains: $$C\text{ov}(Yt, Ys) = \sigma^2 \min(t,s)$$