How to find the median of a p.d.f with unknown integral limits

\begin{align} \int_{s}^t f(x) \, dx &= \int_{s}^t \frac{1}{b}\exp\left(-\frac{x-a}{b}\right) \exp\left(-\exp\left(-\frac{x-a}{b}\right)\right) \, dx \\ &= -\int_{\exp\left(\frac{s-a}{b}\right)}^{\exp\left(\frac{t-a}{b}\right)} e^{u} \, du & u = -\exp\left(-\frac{x-a}{b}\right) \\ &= \exp\left(-\exp\left(\frac{t-a}{b}\right)\right) - \exp\left(-\exp\left(\frac{s-a}{b}\right)\right). \end{align} When $s \to -\infty$ and $t \to \infty$, this tends to $1-0=1$, so the PDF is indeed over the whole real line. It remains to solve for what value of $m$ makes $\int_{-\infty}^m f(x) \, dx = \exp\left(-\exp\left(\frac{m-a}{b}\right)\right)$ equal to $1/2$.