Question #e182a

2 Answers
Konstantinos Michailidis
Nov 17, 2017

Take logarithms on both sides

#x^x=2#

#x*logx=log2#

#e^(logx)*logx=log2#

#x=e^(W(log2))#

#x≈1.55961#

Where #W(x)# is the ProductLog function, defined as the inverse of

the function #f(x)=x*e^x#

Also if you plot the function you will see that there no integer solutions.

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Sean
Nov 17, 2017

Please see below.

Explanation:

Approximate forms
#x = e^(W(2 i pi n + log(2)))#, # i (2 pi n - i log(2))!=0#, #n# element #Z#
#x = e^(W_(-1)(2 i pi n + log(2)))#, # i (2 pi n - i log(2))!=0#, #n# element #Z#
#x = e^(W_1(2 i pi n + log(2)))#, #i (2 pi n - i log(2))!=0#, #n# element #Z#

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