= Solution
{c}
Taking the logarithm of the explicit solution gives a linear function of the <normal random variable> $B_t$:
$$
\log X_t=\log x+(r-\sigma^2/2)t+\sigma B_t.
$$
Thus
$$
\boxed{X_t\sim\operatorname{Lognormal}
(\log x+(r-\sigma^2/2)t,\ \sigma^2t).}
$$
For $z>0$ its density is
$$
\frac1{z\sigma\sqrt{2\pi t}}
\exp\left[-\frac{(\log(z/x)-(r-\sigma^2/2)t)^2}
{2\sigma^2t}\right],
$$
and it is zero for $z\leq0$. The factor $1/z$ is the Jacobian of the logarithmic change of variable.
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