The problem of the quickest detection of a change in the drift of a time-homogeneous diffusion process is considered under the assumption that the detection delay is exponentially penalized. In this framework, the past literature has shown that a two- or three-dimensional optimal stopping problem needs to be faced. In this note, we show how a change of measure significantly simplifies the setting by reducing the dimension of the optimal stopping problem to one or two, respectively. We illustrate this result in the well known Brownian motion case analyzed by Beibel [4] and when a Bessel process is observed, generalizing therefore the results for the linear penalty case obtained by Johnson and Peskir [13].