$\displaystyle f=x^x$

$\displaystyle ln(f)=x*ln(x)$

I am aware of the proof that:

$\displaystyle \frac{d}{dx}a^x=ln(a)*a^x$

but ln(f)=x*ln(x) seems to be following a different format since it would have to be the quantity x^x*ln(x) however my lecture notes seem to be saying that ln(f)=x*ln(x)?

Also with two x's it gets confusing on which one is the exponent and which is the base.

Thanks in advance...