Question

The real values of x satisfy the inequality

${x}^{ log_{3}(x) } > 3$
Quality answer needed.

Moderators, Brainly or genius, please answer it.

Thank you in advance. ​

Answer 1

ANSWER:

$A(n) \in \Theta(n^{\log_b a}) = \Theta(n^{\log_2 2} ) = \Theta(n)$

PLEASE MARK ME AS BRAINLIEST

Answer 2

$\large\underline{\sf{Solution-}}$

Given Logarithmic inequality is

$\rm :\longmapsto\: {x}^{ log_{3}(x) } > 3$

Now, obviously log is defined when x > 0.

So, 2 cases arises,

Case:- 1 When 0 < x < 1

Case :- 2 When x > 1

Consider,

$\rm :\longmapsto\:When \: 0 < x < 1$

$\rm :\implies\:log_{3}(x) < 0$

So,

$\rm :\longmapsto\: {x}^{ log_{3}(x) } > 3$

On taking log on both sides,

$\rm :\longmapsto\: log({x}^{ log_{3}(x) }) > log3$

$\rm :\longmapsto\: log_{3}(x) \: logx > log3$

$\rm :\longmapsto\: log_{3}(x) \: > \dfrac{log3}{logx}$

$\rm :\longmapsto\: log_{3}(x) \: > \dfrac{1}{ log_{3}(x) }$

$\rm :\longmapsto\: {log_{3}(x) } {}^{2} \: > 1$

$\rm :\longmapsto\: {log_{3}(x) } {}^{2} \: - 1 > 0$

$\rm :\longmapsto\: (log_{3}(x) - 1)(log_{3}(x) + 1) > 0$

$\rm :\longmapsto\: log_{3}(x) + 1 < 0 \: \: \: \{ \because \: log_{3}(x) - 1 < 0 \}$

$\rm :\longmapsto\: log_{3}(x) < - \: 1 \:$

$\rm :\longmapsto\:x < {3}^{ - 1}$

$\rm :\implies\:x < \dfrac{1}{3}$

$\bf\implies \:0 < x < \dfrac{1}{3}$

Case :- 2

$\rm :\longmapsto\:When \: x > 1$

$\rm :\implies\:log_{3}(x) > 0$

So, given inequation is

$\rm :\longmapsto\: {x}^{ log_{3}(x) } > 3$

On taking log on both sides, we get

$\rm :\longmapsto\: log({x}^{ log_{3}(x) }) > log3$

$\rm :\longmapsto\: log_{3}(x) \: logx > log3$

$\rm :\longmapsto\: log_{3}(x) \: > \dfrac{log3}{logx}$

$\rm :\longmapsto\: log_{3}(x) \: > \dfrac{1}{ log_{3}(x) }$

$\rm :\longmapsto\: {log_{3}(x) } {}^{2} \: > 1$

$\rm :\longmapsto\: {log_{3}(x) } {}^{2} \: - 1 > 0$

$\rm :\longmapsto\: (log_{3}(x) - 1)(log_{3}(x) + 1) > 0$

$\rm :\longmapsto\: log_{3}(x) - 1 > 0 \: \: \: \{ \because \: log_{3}(x) + 1 > 0 \}$

$\rm :\longmapsto\: log_{3}(x) > \: 1 \:$

$\rm :\longmapsto\:x > {3}^{1}$

$\bf\implies \:x > 3$

Hence,

The real values of x satisfy the

$\rm :\longmapsto\: {x}^{ log_{3}(x) } > 3$

is

$\bf\implies \:\boxed{ \bf \:x \: \in \: \bigg(0, \: \dfrac{1}{3} \bigg) \: \cup \: (3, \: \infty )}$