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Question Number 172744 by Adilali last updated on 30/Jun/22

	Answered by aleks041103 last updated on 30/Jun/22

	$o b v . x = 0 i s a s o l u t i o n .$ $i f x < 0 \to o b v . n o s o l n .$ $i f x > 0 :$ $2^{x} l n (x) = l n (2) + l n (x)$ $\Rightarrow (2^{x} - 1) l n (x) = l n (2)$ $\Rightarrow l n (x) = \frac{l n (2)}{2^{x} - 1}$ $f o r x > 0$ $l n (x) i s s t r i c t l y i n c r e a s i n g$ $\frac{l n (2)}{2^{x} - 1} i s s t r i c t l y d e c r e a s i n g$ $\Rightarrow i n c r . = d e c r .$ $\Rightarrow f o r x > 0, e x i s t a t m o s t 1 s o l u t i o n$ $x = 1 : (2^{x} - 1) l n (x) = 0 < l n (2)$ $x = 2 : (2^{x} - 1) l n (x) = 3 l n (2) > l n (2)$ $\Rightarrow \exists x_{1} \in (1, 2), (2^{x_{1}} - 1) l n (x_{1}) = l n (2)$ $n u m e r i c a l l y :$ $x = 0 a n d x \approx 1.476$
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