 Critical exponential moment of a drifted Brownian hitting time

For $a>0$ and $T_a=\inf \{t\ge 0:B_t+t=a\}$,Changing measure with the density $e^{-B_t-t/2}$ turns $B_t+t$ into Brownian motion. At $T_a$ the density equals $e^{-a+T_a/2}$, and Brownian motion hits $a$ almost surely.