If 2^(x ) and 3^x are integers for some x∈R^+ , must x be an integer?


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Question Number 5185 by Yozzii last updated on 28/Apr/16

$I f 2^{x} a n d 3^{x} a r e i n t e g e r s f o r s o m e$ $x \in R^{+}, m u s t x b e a n i n t e g e r ?$
	Answered by 123456 last updated on 28/Apr/16

	$2^{x} = a \Leftrightarrow x = \log_{2} a$ $3^{x} = b \Leftrightarrow x = \log_{3} b$ $\log_{2} a = \log_{3} b$ $k \in Z$ $a = 2^{\log_{3} b}, a \in Z \Rightarrow \log_{3} b = k \Leftrightarrow b = 3^{k} \Rightarrow x \in Z$ $b = 3^{\log_{2} a}, b \in Z \Rightarrow \log_{2} a = k \Leftrightarrow a = 2^{k} \Rightarrow x \in Z$
	Commented by Yozzii last updated on 28/Apr/16

	$I n l i n e 5, a \in Z I t h i n k c o u l d a l s o$ $m e a n 2^{{l o g}_{3} b} = 2^{{l o g}_{2} n} = n f o r s o m e n \in Z .$ $\Rightarrow b = 3^{{l o g}_{2} n}$ $B u t b, n \in Z^{+} \Rightarrow n = 2^{j} \in Z \Rightarrow x = j \in Z$ $3^{{l o g}_{2} a} = 3^{{l o g}_{3} q} \Rightarrow a = 2^{{l o g}_{3} q}$ $\Rightarrow q = 3^{r} \Rightarrow x = r \in Z^{+} .$
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