If-2-x-and-3-x-are-integers-for-some-x-R-must-x-be-an-integer-

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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