A student claims that both ln(ex)=x and eln x=x hold for every real x. Give the exact domain of vali

Question

A student claims that both ln(ex)=x and eln x=x hold for every real x. Give the exact domain of validity of each identity and use x=-1 to explain the difference.

Accepted Answer

First identity: $x\in\mathbb R$. Second identity: $x>0$. At $x=-1$, only the first is defined.

Comparing two inverse compositions

Comparing two inverse compositions

Solution:

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