$2^x=x^2$

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photon
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$2^x=x^2$

Unread post by photon » Mon Feb 06, 2012 2:34 pm

find x integer, if $2^x=x^2$(just made a change)..
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Phlembac Adib Hasan
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Re: $2^x=x^2$

Unread post by Phlembac Adib Hasan » Mon Feb 06, 2012 2:53 pm

My solve :
\[2^x=x^2\]
\[\Rightarrow x=2.log_2 x\]
LHS is a integer.So $x$ must be a power of $2$.
Let $x=2^k$
So the equation becomes \[2^{2^k}=2^{2k}\]
\[\Rightarrow 2^k=2k\]
\[\Rightarrow k=2^{k-1}\]
It's easy to notice $k=1,2$.So only solutions are $x=(2,4)$
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shehab ahmed
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Re: $2^x=x^2$

Unread post by shehab ahmed » Wed Mar 21, 2012 2:56 pm

এরকম আরেকটি সমস্যা হল $x^y=y^x$.এক্ষেত্রেও x,y এর কেবল একটি পূর্ণসাংখ্যিক সমাধান আছে।তা হল x=2,y=4 আর উল্টাটা।
Last edited by *Mahi* on Wed Mar 21, 2012 6:03 pm, edited 1 time in total.
Reason: LaTeXed
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