Question

Calcule logaritmo de 216 na base 54, sabendo que logaritmo na base 10 de 8 vale "a" e que logaritmo na base 10 de 9 vale "b"

Answer 1

$a=log8\\a=log2^3\\a=3log2\\log2=\frac{a}{3}\\\\b=log9\\b=log3^2\\b=2log3\\log3=\frac{b}{2}\\\\\\E=log_{54}216\\\\E=\frac{log216}{log54}\\\\E=\frac{log(2^3*3^3)}{log{(2*3^3)}}\\\\E=\frac{log2^3+log3^3}{log2+log3^3}\\\\E=\frac{3log2+3log3}{log2+3log3}\\\\E=\frac{3*\frac{a}{3}+3*\frac{b}{2}}{\frac{a}{3}+3*\frac{b}{2}}\\\\E=\frac{a+\frac{3b}{2}}{\frac{a}{3}+\frac{3b}{2}}\\\\E=\frac{\frac{2a+3b}{2}}{\frac{2a+9b}{6}}\\\\E=\frac{6*(2a+3b)}{2*(2a+9b)}\\\\E=\frac{3*(2a+3b)}{2a+9b}\\\\\boxed{E=\frac{6a+9b}{2a+9b}}$

Dúvidas? Comente.

Answer 2

A solução está no anexo