Is this power rule true for the natural base?

Question

Two questions

1) I was wondering if $e^{k \ln{x}}=k$ for any k. Is it?
2)To test I went to Maple and typed e^-ln(x) and it gave $e^{-ln(x)}$. I tried simplify and it gave $x^{-ln e}$. How do you get Maple to simplify properly? Why doesn't it evaluate ln e = 1 ?

Answer 1

To Maple "e" is just a letter, it does not recognise it as $e$. Try "exp(-ln(x))".

Also (to make the point clear) try "evalf(e)" then "e:=exp(1)" and then "evalf(e)" again.

Answer 2

Yes, it is true for any complex number k.

That's why Maple does the following simplification.

simplify( exp(k*ln(x)) );

                                  k
                                 x