Question

JEE MAINS MATHS QUESTION SEPTEMBER 2020

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

$log_{ {7}^{ \frac{1}{2} } }x + log_{ {7}^{ \frac{1}{3} } }x + log_{ {7}^{ \frac{1}{4} } }x + ......20 \: times = 460$

$\implies \displaystyle \: \: \frac{1}{ log_{x}( {7}^{ \frac{1}{2} } ) } + \frac{1}{ log_{x}( {7}^{ \frac{1}{3} } ) } + \frac{1}{ log_{x}( {7}^{ \frac{1}{3} } ) } + .....20 \: \: times \: = 460$

$\implies \displaystyle \: \: \frac{1}{ \frac{1}{2} log_{x}7 } + \frac{1}{ \frac{1}{3} log_{x}7 } + \frac{1}{ \frac{1}{4} log_{x}7 } + ....20 \: times \: = 460$

$\implies \displaystyle \: \: \frac{2}{ log_{x}7 } + \frac{3}{ log_{x}7 } + \frac{4}{ log_{x}7 } + ...20 \: times \: = 460$

$\implies \displaystyle \: \: 2 \: log_{7}x + 3 \: log_{7}x + 4 \: log_{7}x + .....20 \: times \: = 460$

$\implies \displaystyle \: \: log_{7}x \times (2 + 3 + 4 + ....20 \: times \: ) = 460$

$\implies \displaystyle \: \: log_{7}x \times 230 = 460$

$\implies \displaystyle \: \: log_{7}x = 2$

$\implies \displaystyle \: x = {7}^{2}$

$\therefore \: x = 49$