Lost Logarithm II

Algebra Level 3

Which of the following relation is true for positive real numbers $a$ , $b$ and $x$

A : $\log_{ab} x =\dfrac{\log_x a\log_x b}{\log_xa+\log_xb}$

B : $\log_{ab} x =\dfrac{\log_a x\log_b x}{\log_xa+\log_xa}$

C : $\log_{ab} x =\dfrac{\log_x a\log_x b}{\log_ax+\log_bx}$

D : $\log_{ab} x =\dfrac{\log_a x\log_b x}{\log_ax+\log_bx}$

D A B C

1 solution

Danish Ahmed
Feb 28, 2015

$\dfrac{\log_{a}x\log_{b}x}{\log_{a}x+\log_{b}x}$

$=\dfrac{\dfrac{\log x}{\log a}\dfrac{\log x}{\log b}}{\dfrac{\log x}{\log a}+\dfrac{\log x}{\log b}}$

$=\dfrac{\dfrac{\log x}{\log a\log b}}{\dfrac{\log a+\log b}{\log a\log b}}=\dfrac{\log x}{\log a+\log b}=\dfrac{\log x}{\log ab}=\log_{ab}x$