# CAT 2020 Quant Question [Slot 1] with Solution 08

Question

If y is a negative number such that ${2^{{y^2}{{\log }_3}5}} = {5^{{{\log }_2}3}}$, then y equals

1. ${\log _2}(1/3)$
2. $- {\log _2}(1/3)$
3. ${\log _2}(1/5)$
4. $- {\log _2}(1/5)$
Option: 1
Solution:

${2^{{y^2}{{\log }_3}5}} = {5^{{{\log }_2}3}}$

${\left( {{2^{{{\log }_3}5}}} \right)^{{y^2}}} = {5^{{{\log }_2}3}}$

${\left( {{5^{{{\log }_3}2}}} \right)^{{y^2}}} = {5^{{{\log }_2}3}}$

${5^{{y^2}{{\log }_3}2}} = {5^{{{\log }_2}3}}$

$\Rightarrow {y^2}{\log _3}2 = {\log _2}3$

${y^2} = \left( {{{\log }_2}3} \right)\left( {{{\log }_2}3} \right)$

is negative )

$y = {\log _2}{3^{ - 1}} = {\log _2}\frac{1}{3}$

CAT 2021 Online Course @ INR 8999 only