- #1

chwala

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- Homework Statement:
- ##2^x##=##x^2##

- Relevant Equations:
- trial/error method

ok, by direct substitution i know that either ##x=2## or ##x=4##

but i would like to prove this analytically, would it be correct saying,

##xln 2= 2ln x##

##xln_{2}2=2 ln_{2}x##

##x=2 ln_{2}x##

##\frac {1}{2}=\frac { ln_{2}x}{x}##

##ln_{2}x^{1/x}##=##\frac {1}{2}##

→##2^{1/2}##=##x^{1/x}##

→##x=2##

is my thinking correct? and how do we show the solution ##x=4?## analytically

but i would like to prove this analytically, would it be correct saying,

##xln 2= 2ln x##

##xln_{2}2=2 ln_{2}x##

##x=2 ln_{2}x##

##\frac {1}{2}=\frac { ln_{2}x}{x}##

##ln_{2}x^{1/x}##=##\frac {1}{2}##

→##2^{1/2}##=##x^{1/x}##

→##x=2##

is my thinking correct? and how do we show the solution ##x=4?## analytically