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## why is x^(1/x) = e^((1/x)lnx)?

lucas7,

The Natural logarithm function is the INVERSE of the exponential function (for base, e). That is why the method works.

How do you start with a number, x, put into a function, and then put this function into another function, and the outcome be x? One function undoes the effect of the other function.

y=e^x
y is the value along the vertical number line, x is the value along the horizontal number line. What if you switch x and y?
x=e^y
What function is this? What is "y"? We call this the natural logarithm function,
y=ln(x), and this is the inverse of y=e^x.

Pick an x, any x. e^(ln(x))=x, and ln(e^x))=x
 $$e^{ln(x)}=x$$ because they are inverse functions. $$f(f^{-1}(x))=x$$ by definition
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