= Solution
{c}
Under the <risk-neutral measure>, set $a=\sigma\sqrt\tau$ and $m=\log(S/K)+(r-\sigma^2/2)\tau$. Then $\log(S_T/K)=m+aZ$ with $Z$ a <standard normal random variable>, and $d_-=m/a$. The <log-call option> is the discounted <Gaussian positive-part expectation>:
$$
\begin{aligned}
V(S,t)&=e^{-r\tau}\int_{-d_-}^{\infty}(m+az)\phi(z)\,dz\\
&=e^{-r\tau}\bigl[m\Phi(d_-)+a\phi(d_-)\bigr],
\end{aligned}
$$
since $\phi'(z)=-z\phi(z)$. Therefore
$$
\boxed{V(S,t)=e^{-r\tau}\sigma\sqrt\tau
\left[d_-\operatorname N(d_-)+\frac{e^{-d_-^2/2}}{\sqrt{2\pi}}\right].}
$$
Here $\phi$ and $\Phi=\operatorname N$ are the <standard normal density> and <standard normal distribution function>. At expiry the continuous terminal <limit> is $(\log(S/K))^+$.
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