### 7.300 Which number is greater: $\log _{135} 675$ or $\log _{45} 75$ ?
Solution. $\log _{135} 675=\frac{\log _{3} 675}{\log _{3} 135}=\frac{\log _{3}(9 \cdot 75)}{\log _{3}(3 \cdot 45)}=\frac{2+\log _{3} 75}{1+\log _{3} 45}=\frac{\left(2+\log _{3} 75\right) \log _{3} 45}{\left(1+\log _{3} 45\right) \log _{3} 45}$.

$
\log _{45} 75=\frac{\log _{3} 75}{\log _{3} 45}=\frac{\log _{3} 75 \cdot\left(1+\log _{3} 45\right)}{\log _{3} 45 \cdot\left(1+\log _{3} 45\right)}
$

But $2 \log _{3} 45+\log _{3} 75 \cdot \log _{3} 45>\log _{3} 75+\log _{3} 75 \cdot \log _{3} 45 \Rightarrow$

$\Rightarrow \log _{135} 675>\log _{45} 75$.

Answer: the number $\log _{135} 675$ is greater.