A complete Bernstein function related to the fractal dimension of Pascal's pyramid modulo a prime

Let $f_r(x)=\log (1+rx)\mathrm{/}\log (1+x)$ for $x>0$. We prove that $f_r$ is a complete Bernstein function for $0\le r\le 1$ and a Stieltjes function for $1\le r$. This answers a conjecture of David Bradley that $f_r$ is a Bernstein function when $0\le r\le 1$.