 Solution

Subtract the reflected solution from part c. On the alive side of the down-and-out barrier option,Both terms satisfy the Black-Scholes equation, and at $S=B$ they cancel. At expiry, $S\ge B$ gives $B^2/S\le B<K$, so the reflected log-call option has zero terminal financial payoff; the first term gives the required payoff. With the usual growth condition, uniqueness for the killed pricing problem identifies the expression as the risk-neutral price. After the barrier has been hit the contract value is zero. The algebraic image expression on $S<B$ is not its financial value.