\begin{multline*}
      \lxtoo\frac{\ln\sin ax}{\ln\sin bx}\lh
      \lxtoo\frac{a\cos ax\cdot\frac1{\sin ax}}
                 {b\cos bx\cdot\frac1{\sin bx}}=
      \lxtoo\frac{a\sin bx}{b\sin ax}=\frac ab\lxtoo\frac{\sin bx}{\sin ax}\lh\\
      \lh\frac ab\lxtoo\frac{b\cos bx}{a\cos ax}=\frac ab\cdot \frac ba=1.
   \end{multline*}