Solving log(x)=x−1? One can use Taylor series of the log or exp function to get the result that x=1. I was wondering if there is any other simple solutions.

easternerjx

easternerjx

Answered question

2022-09-23

Solving log ( x ) = x 1?
One can use Taylor series of the log or exp function to get the result that x = 1. I was wondering if there is any other simple solutions.
Thanks a lot!

Answer & Explanation

unfideneigreewl

unfideneigreewl

Beginner2022-09-24Added 5 answers

The function f ( x ) = x log x has derivative f ( x ) = 1 1 x which is negative for x < 1 and positive for x > 1. Therefore, f is strictly decreasing for x < 1 and strictly increasing for x > 1. Since f ( 1 ) = 1 we have f ( x ) > 1 for x < 1 and x > 1. Therefore, x = 1 is the only solution to f ( x ) = 1
Marcelo Maxwell

Marcelo Maxwell

Beginner2022-09-25Added 2 answers

There is also an infinite set of complex solutions (if you already had complex numbers) if you consider different branches of the complex logarithm (Where k denotes the branch):
log ( k , x ) = x 1
x = e x 1
x e 1 x = 1
x e x = 1 e
x = W k ( 1 e ) k Z
where W k is the k-th branch of the W-Function.
Note that if you are taking the "main" branch of the logarithm in your equation, there is only one solution: x = 1

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