2^x −3^(((x/2))) =1


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Question Number 148048 by puissant last updated on 25/Jul/21

$2^{x} - 3^{(\frac{x}{2})} = 1$
	Answered by mindispower last updated on 25/Jul/21

	$2^{x} - {(\sqrt{3})}^{x} = 1$ $\Leftrightarrow 1 = {(\frac{1}{2})}^{x} + {(\frac{\sqrt{3}}{2})}^{x}$ $x = 2 s o l u t i o n$ $i f x ⩽ 0, {(\frac{1}{2})}^{x} + {(\frac{\sqrt{3}}{2})}^{x} ⩾ 2$ $x \to {(\frac{1}{2})}^{x} + {(\frac{\sqrt{3}}{2})}^{x} d e c r e a s e f u n c t i o n f o r x > 0$ $f (2) = 1, \Rightarrow x = 2$
	Answered by Rasheed.Sindhi last updated on 25/Jul/21

	$2^{x} - 3^{(\frac{x}{2})} = 1$ $2^{2 y} - 3^{y} = 1 [x / 2 \to y]$ $4^{y} - 3^{y} = 1$ $y > 1 \Rightarrow 4^{y} - 3^{y} > 1$ $y > 1 \Rightarrow x / 2 > 1 \to N o s o l u t i o n$ $y = 1 \Rightarrow x = 2$
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