Let m be the minimum possible value of log 3 (3 y 1 +3 y 2 +3 y 3 ) , where y 1 , y 2 , y 3 are real numbers for which y 1 +y 2 +y 3 =9 . Let M be the maximum possible value of (log 3 x 1 +log 3 x 2 +log 3 x 3 ) , where x 1 , x 2 , x 3 are positive real numbers for which x 1 +x 2 +x 3 =9 . Then the value of log 2 (m 3 )+log 3 (M 2 ) is

Mathematics · Logrithms

JEE Advanced 2020 — Paper 1 — Question 40

Let m be the minimum possible value of $\log _{3}\left(3^{y_{1}}+3^{y_{2}}+3^{y_{3}}\right)$ , where $y_{1}, y_{2}, y_{3}$ are real numbers for

which $y_{1}+y_{2}+y_{3}=9$ . Let $M$ be the maximum possible value of $\left(\log _{3} x_{1}+\log _{3} x_{2}+\log _{3} x_{3}\right)$ ,

where $x_{1}$ , $x_{2}, x_{3}$ are positive real numbers for which $x_{1}+x_{2}+x_{3}=9$ . Then the value of

$\log _{2}\left(m^{3}\right)+\log _{3}\left(M^{2}\right)$ is

Numerical answer — enter this value.

$\frac{3^{y_{1}}+3^{y_{2}}+3^{y_{3}}}{3} \geq 3^{\frac{y_{1}+y_{2}+y_{3}}{2}} \quad($ Using $A M \geq G M)$

$\Rightarrow \log _{3}\left(3^{y_{1}}+3^{y_{2}}+3^{y_{3}}\right) \geq \log _{3} 3^{4}$ $\geq 4$

$\Rightarrow \mathrm{m}=4$

Similarly, $\frac{x_{1}+x_{2}+x_{3}}{3} \geq\left(x_{1} x_{2} x_{3}\right)^{\frac{1}{3}}$

$\log _{3} 3 \geq \frac{1}{3}\left(\log _{3} x_{1}+\log _{3} x_{2}+\log _{3} x_{3}\right)$

$3 \geq \log _{3} x_{1}+\log _{3} x_{2}+\log _{3} x_{3}$

$\Rightarrow M=3$

$\log _{2} m^{3}+\log _{3} M^{2}=8$

Answer key and solution verified before publishing.

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