Let $Y = \log X$, then:

$$\begin{align} Y &= Y*0 + (\mu-\frac{\sigma^2}{2})t + \sigma W*t \\ EY*t &=Y*0 + (\mu-\frac{\sigma^2}{2})t \\ EY*tEY*s &= Y*0^2 + Y*0 (\mu-\frac{\sigma^2}{2}) (t+s) + (\mu-\frac{\sigma^2}{2})^2 t s \\ E(Y*tY*s) &= E\left((Y*0 + (\mu-\frac{\sigma^2}{2})t + \sigma W*t) (Y*0 + (\mu-\frac{\sigma^2}{2})s + \sigma W*s)\right) \\ &= Y*0^2 + Y*0 (\mu-\frac{\sigma^2}{2}) (t+s) + (\mu-\frac{\sigma^2}{2})^2 t s + \dots + \sigma ^2 \min(t,s) \end{align}$$

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