Question

If the root of $4^{\log _{10}(10 x)}-6^{\log _{10} x}=2 \cdot 3^{\log _{10}\left(100 x^2\right)}$ is $x=\frac{p}{q}$, where $p \& q$ are coprime natural numbers then $p + q =$

• A. 99
• B. 100
• C. 101
• D. 102

Answer 1

Answer: (C)

$a=2^{\log x} \text { and } b=3^{\log x}$
$\therefore 4 a^2-a b-18 b^2=0 $
$\therefore \frac{a}{b}=\left(\frac{2}{3}\right)^{\log x}=\frac{9}{4}=\left(\frac{2}{3}\right)^{-2} $
$\log _{10} x=-2 \rightarrow x=\frac{1}{100}$