= Solution
For radial <null geodesics>, the metric gives $dt/dr=\pm f^{-1}$ in the exterior. Define the <tortoise coordinate> by
$$
\boxed{r_\star(r)=\int^r\frac{dx}{f(x)}.}
$$
An outgoing future radial ray has $dt/dr=f^{-1}$, so $u=t-r_\star$ is constant; an ingoing ray has $dt/dr=-f^{-1}$, so $v=t+r_\star$ is constant. These describe the null trajectories independently of the choice of <affine parameter>.
For the <tortoise coordinate of a five-dimensional charged black hole>, with $a=r_+$, $b=r_-$ and $\Delta=a^2-b^2$, partial fractions give
$$
\frac1{f(r)}=1+\frac{a^4}{\Delta(r^2-a^2)}-\frac{b^4}{\Delta(r^2-b^2)}.
$$
Thus a useful explicit <tortoise coordinate> is
$$
\boxed{r_\star=r+\frac{a^3}{2\Delta}\log\left|\frac{r-a}{r+a}\right|-\frac{b^3}{2\Delta}\log\left|\frac{r-b}{r+b}\right|+C.}
$$
The logarithms signal failure of the static coordinates at the horizons, rather than a curvature singularity there.
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