Question

If $b^2=ac$ and $a,b,c$ are positive numbers, then the value of $\frac{1}{{\log }_{a}x}+\frac{1}{{\log }_{c}x}$ will be

• A. $\frac{1}{{\log }_{b}x}$
• B. $\frac{2}{{\log }_{c}x}$
• C. $\frac{2}{{\log }_{b}x}$
• D. $\frac{1}{{\log }_{c}x}$

Answer 1

Answer: (C)

$\frac{2}{{\log }_{b}x}$

$b^2=ac\Rightarrow {\log }_x{b}^2={\log }_{x}ac$

$\Rightarrow 2{\log }_{x}b={\log }_{x}a+{\log }_{x}c[\because \log (ab)=\log a+\log b]$

$\Rightarrow \frac{2}{{\log }_{b}x}=\frac{1}{{\log }_{a}x}+\frac{1}{{\log }_{c}x}[\because {\log }_{b}a=\frac{1}{{\log }_{a}b}]$