The value of x in the given equation 4^x −3^(x−(1/2)) = 3^(x+(1/2)) −2^(2x−1) , is


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Question Number 114551 by bemath last updated on 19/Sep/20

$T h e v a l u e o f x i n t h e g i v e n e q u a t i o n$ $4^{x} - 3^{x - \frac{1}{2}} = 3^{x + \frac{1}{2}} - 2^{2 x - 1}, i s_,$
Commented by Dwaipayan Shikari last updated on 19/Sep/20

$2^{2 x} + 2^{2 x - 1} = 3^{x + \frac{1}{2}} + 3^{x - \frac{1}{2}}$ $2^{2 x} (1 + \frac{1}{2}) = 3^{x - \frac{1}{2}} (3 + 1)$ $2^{2 x - 1} . 3 = 3^{x - \frac{1}{2}} . 2^{2}$ $2^{2 x - 3} = 3^{\frac{2 x - 3}{2}}$ $(2 x - 3) l o g 2 = \frac{2 x - 3}{2} l o g (3)$ $(2 x - 3) (l o g 2 - \frac{1}{2} l o g 3) = 0$ $2 x - 3 = 0$ $x = \frac{3}{2}$
Commented by Rasheed.Sindhi last updated on 19/Sep/20

$A l t e r n a t e w a y (t o p r e v e n t l o g)$ $F r o m a b o v e :$ $2^{2 x - 3} = 3^{\frac{2 x - 3}{2}}$ ${(\frac{2}{\sqrt{3}})}^{2 x - 3} = {(\frac{2}{\sqrt{3}})}^{0}$ $2 x - 3 = 0$ $x = \frac{3}{2}$
Commented by Dwaipayan Shikari last updated on 19/Sep/20

$Y e s s i r!$
	Answered by bobhans last updated on 19/Sep/20

	$\Leftrightarrow 4^{x} + \frac{1}{2} . 4^{x} = (\sqrt{3} + \frac{1}{\sqrt{3}}) . 3^{x}$ $\Leftrightarrow \frac{3}{2} . 4^{x} = \frac{4}{3} \sqrt{3} . 3^{x}$ $\Leftrightarrow 2^{2 x - 3} = 3^{x - \frac{3}{2}} \Rightarrow (2 x - 3) \ln (2) = (x - \frac{3}{2}) \ln (3)$ $\Rightarrow x (2 \ln 2 - \ln 3) = 3 \ln 2 - \frac{3}{2} \ln 3$ $\Rightarrow x = \frac{1}{2} (\frac{6 \ln 2 - 3 \ln 3}{2 \ln 2 - \ln 3}) = \frac{3}{2} (\frac{2 \ln 2 - \ln 3}{2 \ln 2 - \ln 3})$ $\Rightarrow x = \frac{3}{2}$
	Commented by bemath last updated on 19/Sep/20

	$g a v e k u d o s$
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