Question

In section 90, the professor explains that
X^X = e ^ xlnx. Could some one explains little details about this and how is it possible to write in this way

Answer 1

by definition,
ln x = some exponent (call it "a") such that e^a = x

that is, given
ln x = a

e^a = x
or using a = ln x
\[ e^{\ln x} = x\]
this is an identity

Answer 2

if we write
\[ x = e^{\ln x} \]
then
\[ x^x = \left( e^{\ln x} \right)^x \]

there is a property of exponents:
\[ \left(a^b\right)^c = a^{b c}\]
which allows us to write
\[ x^x = \left( e^{\ln x} \right)^x = e^{x\ln x} \]