x^2 log_3 x^2 −(2x^2 +3)log_9 (2x+3)=3log


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Question Number 204300 by depressiveshrek last updated on 11/Feb/24

$x^{2} \log_{3} x^{2} - (2 x^{2} + 3) \log_{9} (2 x + 3) = {3 \log}_{3} (\frac{x}{2 x + 3})$
	Answered by Rasheed.Sindhi last updated on 11/Feb/24

	$x^{2} \log_{3} x^{2} - (2 x^{2} + 3) \log_{9} (2 x + 3) = {3 \log}_{3} (\frac{x}{2 x + 3})$ $\log_{9} (2 x + 3) = \frac{\log_{3} (2 x + 3)}{\log_{3} 9} = \frac{\log_{3} (2 x + 3)}{2}$ $2 x^{2} \log_{3} x - (2 x^{2} + 3) (\frac{\log_{3} (2 x + 3)}{2}) = {3 \log}_{3} (\frac{x}{2 x + 3})$ $2 x^{2} \log_{3} x - (2 x^{2} + 3) \log_{3} (2 x + 3) = {6 \log}_{3} (\frac{x}{2 x + 3})$ $4 x^{2} \log_{3} x - (2 x^{2} + 3) \log_{3} (2 x + 3) = \log_{3} {(\frac{x}{2 x + 3})}^{6}$ $\log_{3} x^{4 x^{2}} - \log_{3} {(2 x + 3)}^{2 x^{2} + 3} = \log_{3} {(\frac{x}{2 x + 3})}^{6}$ $\log_{3} (x^{4 x^{2}} \div {(2 x + 3)}^{2 x^{2} + 3}) = \log_{3} {(\frac{x}{2 x + 3})}^{6}$ $x^{4 x^{2}} \div {(2 x + 3)}^{2 x^{2} + 3} = {(\frac{x}{2 x + 3})}^{6}$ $\frac{x^{4 x^{2}}}{{(2 x + 3)}^{2 x^{2} + 3}} = {(\frac{x}{2 x + 3})}^{6}$
	Commented by deleteduser1 last updated on 12/Feb/24

	$\Rightarrow x^{4 x^{2} - 6} = {(2 x + 3)}^{2 x^{2} - 3} \Rightarrow {(x^{2})}^{2 x^{2} - 3} = {(2 x + 3)}^{2 x^{2} - 3}$ $\Rightarrow {(\frac{x^{2}}{2 x + 3})}^{2 x^{2} - 3} = 1 \Rightarrow \frac{x^{2}}{2 x + 3} = 1 \lor 2 x^{2} - 3 = 0$ $\Rightarrow x = 3, x = \frac{\sqrt{6}}{2},$
	Commented by Rasheed.Sindhi last updated on 12/Feb/24

	$T h a n k s s i r!$
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