Выразите log_4(500) через a и b, если log_3(20)=a и log_3(5)=b.

Решите задачу:

$\log_4500=\frac{\log_3{500}}{\log_3{4}}=\frac{\log_3{(125\cdot4)}}{\log_3{4}}=\\ =\frac{\log_3{125}+\log_3{4}}{\log_3{4}}=\frac{\log_3{5^3}+\log_3{4}}{\log_3{4}}=\\ \frac{3\log_3{5}}{\log_3{4}}+1=\frac{3\log_3{5}}{\log_3{\frac{20}{5}}}}+1=\\ \frac{3\log_3{5}}{\log_3{20}-\log_3{5}}+1=\frac{3b}{a-b}+1=\\ \frac{a+2b}{a-b}$