why is that so?

Question

e^lnx=x

i understand why ln(e^x)=x, but not the other way around

Answer 1

Consider this: let y = ln x

Then e^y = x

If we ln both sides,
ln(e^y) = ln x
y = lnx
ln x = ln x

Thus e^ln(x) = x


Alternatively, if we define ln(x) = y, then

x = e^y (be definition of a logarithm)
x = e^ln(x) (because y = ln(x))

Answer 2

If you understand that ln(e^x)=x you can realize that taking the ln of the left side of [e^lnx=x] gives you [ln(e^lnx)=lnx]

Answer 3

1_(e^x)=T so 2_ ln(e^x)=lnT so 3_ x=lnT from 1 and 3 e^lnT=T. Good luck