Let $a$ and $b$ be positive reals both greater than $1$. I'd like to compute the limit of the summation \begin{eqnarray} \lim_{x \to \infty} \frac{1}{\log_a x} \sum_{k = 0}^{\lfloor \log_{a} x \rfloor} \left \lbrace (\log_{b} x) - k \log_b a \right \rbrace, \end{eqnarray} where $\{ \cdot \}$ denotes the fractional part function and $\lfloor \cdot \rfloor$ denotes the floor function. I believe that the limit is $\frac{1}{2}$ for almost all $a$ and $b$, and I believe the limit doesn't exist if $b = a^r$ where $r$ is a positive rational. I also have a feeling that Weyl's theorem may be playing a role here but I'm not sure if $\lbrace (\log_{b} x) - k \log_b a \rbrace$ is equidistributed on any interval.
Any help is certainly appreciated!