Rewriting the function e^-x*x^t-1

[neptune12XII]

does e^-x*x^(t-1)=
e^(t*ln(x)-ln(x)-x)
heres my reasoning:
x=e^ln(x)
e^-x*x^(t-1)=
e^-x*e^(ln(x)(t-1))=
e^-x*e^(t*ln(x)-ln(x))=
e^(t*ln(x)-ln(x)-x)

I want it in the latter form so that it is easier to take derivatives and antiderivatives. did i make any mistakes?

 

[D H]

Where are your parentheses? As written, your expression is $e^{-x}x^t-1 = e^t \ln x - \lnx -x$, which obviously isn't true.

 

[neptune12XII]

youre right srry

 

[HallsofIvy]

If you mean $e^{-x}x^{t-1}$ then it is equal to $e^{-x}e^{ln(x^{t-1})}=$$e^{-x}e^{(t- 1)ln(x)} = e^{-x+ tln(x)- ln(x)}$