  Let $f_r(x)=\log(1+rx)/\log(1+x)$ 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 $0\le r\le 1$.
