log₃ 512 log₄3

log₃8 log₄9

$$\begin{gathered} {\log }_{3}512.{\log }_{4}3 \\ =\frac{{\log }_{2}512}{{\log }_{2}3}\times \frac{{\log }_{2}3}{{\log }_{2}4} \end{gathered}$$

$$\begin{gathered} =\frac{{\log }_{2}512}{{\log }_{2}4} \\ =\frac{9{\log }_{2}2}{2{\log }_{2}2} \\ =\frac{9}{2} \end{gathered}$$

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$$\begin{gathered} {\log }_{3}8.{\log }_{4}9 \\ =\frac{{\log }_{2}8}{{\log }_{2}3}\times \frac{{\log }_{2}9}{{\log }_{2}4} \end{gathered}$$

$$\begin{gathered} =\frac{{\log }_2\left(2^3\right)}{{\log }_{2}3}\times \frac{{\log }_2\left(3^2\right)}{{\log }_2\left(2^2\right)} \\ =\frac{3{\log }_{2}2}{{\log }_{2}3}\times \frac{2{\log }_{2}3}{2{\log }_{2}2} \\ =3 \end{gathered}$$