If $\langle x\rangle $ is the fractional part of $x$, it is known that for $0<\mu<1$, the sequence $\langle \log_\mu n\rangle _{n=1}^\infty$ is dense in $[0,1]$ but is not uniformly distributed. Is this a well studied sequence? Is there any result about the distribution of it?

It doesn't really depend very much on $\mu$, does it, since $\log_{\mu}n=\log n/\log\mu$. I'm not sure there's much to say about the distribution of the fractional part of $\log n$. It is discussed in the Kuipers-Niederreiter book.