1. Determine which number is greater $\log _{3} 4$ or $\log _{4} 5$.
Solution. From the inequality of means, it follows that

$$
\log _{4} 5 \cdot \log _{4} 3 \leq\left(\frac{\log _{4} 5+\log _{4} 3}{2}\right)^{2}=\left(\frac{\log _{4} 15}{2}\right)^{2}<1
$$

therefore

$$
\left.\log _{4} 5<\frac{1}{\log _{4} 3}\right] \log _{3} 4
$$