Logarithm proof fallacious or not? e^(−x)=e^(1/x) Taking the natural logarithm of both sides

Chloe Arnold

Chloe Arnold

Answered question

2022-10-12

Logarithm proof fallacious or not?
e x = e 1 / x Taking the natural logarithm of both sides
ln ( e x ) = ln ( e 1 / x )
x = 1 / x
x 2 = 1
x 2 = 1
x = i
I know I am doing something wrong here. Also can someone please explain why
ln ( x ) = ln ( 1 / x )
Thank you.

Answer & Explanation

vacchetta7k

vacchetta7k

Beginner2022-10-13Added 6 answers

First statement is wrong.
e x = 1 e x e 1 / x
As for the other question, note that if you exponentiate both sides, you get
e ln x = 1 e ln x = 1 x
on the left and the same thing on the left. Certainly in real numbers, x = y iff e x = e y
erwachsenc6

erwachsenc6

Beginner2022-10-14Added 3 answers

It's true that e x = 1 / ( e x ), but e x = e 1 / x is false, hence the 'proof' is meaningless.

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