A Nice Olympiad Exponential Challenge - What is the value of x?

Math Olympiad Question. 2^x=x^32
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[@uwose]

x^32 seems like a very big number. Trying to make 2^x similar big would require a large value for x. That makes x^32 even bigger... So I intuitively expected x to be quite "small"... I love the way to solve this!

[@RealQuInnMallory]

(x ➖ 2x+2)