  We investigate Benford's law in relation to fractal geometry. Basic fractals,
such as the Cantor set and Sierpinski triangle are obtained as the limit of
iterative sets, and the unique measures of their components follow a geometric
distribution, which is Benford in most bases. Building on this intuition, we
aim to study this distribution in more complicated fractals. We examine the
Laurent coefficients of a Riemann mapping and the Taylor coefficients of its
reciprocal function from the exterior of the Mandelbrot set to the complement
of the unit disk. These coefficients are 2-adic rational numbers, and through
statistical testing, we demonstrate that the numerators and denominators are a
good fit for Benford's law. We offer additional conjectures and observations
about these coefficients. In particular, we highlight certain arithmetic
subsequences related to the coefficients' denominators, provide an estimate for
their slope, and describe efficient methods to compute them.
