  We show how rational function approximations to the logarithm, such as $\log
z \approx (z^2 - 1)/(z^2 + 6z + 1)$, can be turned into fast algorithms for
approximating the determinant of a very large matrix. We empirically
demonstrate that when combined with a good preconditioner, the third order
rational function approximation offers a very good trade-off between speed and
accuracy when measured on matrices coming from Mat\'ern-$5/2$ and radial basis
function Gaussian process kernels. In particular, it is significantly more
accurate on those matrices than the state-of-the-art stochastic Lanczos
quadrature method for approximating determinants while running at about the
same speed.
