Why is ln(x^x)=xln(x) valid?



Answered question


Why is ln ( x x ) = x ln ( x ) valid?
I know that ln ( x k ) = k ln ( x ) for any constant k, but why is ln ( x x ) = x ln ( x ). The exponent x is not constant.

Answer & Explanation

Collin Gilbert

Collin Gilbert

Beginner2022-09-09Added 11 answers

As x is probably not an integer, x x is defined as :
x x = e x ln ( x )
Hence, taking the logarithm give you ln x x = x ln ( x )
Janiah Parks

Janiah Parks

Beginner2022-09-10Added 1 answers

another way to think about it, for positive real x , y
(1) ln y = log x y ln x
and, again by definition
log x x x = x

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