Suppose we have a process $X$ with $dX_t=\sigma dB_t + \mu dt$, for constants $\sigma$ and $\mu$, started at $x\in (a,b)$, for some constants $a$ and $b$, where $B$ is a Brownian motion. We'd like to determine the probability that $X$ exits $[a,b]$ via $a$. This can be done by finding the scale function (as explained here, for example).
Is it also possible to solve this problem by applying Girsanov's Theorem? For similar problems, this approach tends to be pretty effective, as it let's us reformulate the problem as one involving hitting times of a Brownian motion, rather than a diffusion.