does ln 1/e^6 = -6
convert into 1/e^6 = e ^-6 ???
In general, ln(a)= b is the same as (converts into) $\displaystyle e^b= a$ so, in particular, $\displaystyle ln(1/e^{6})= -6$, $\displaystyle 1/e^6= e^{-6}$ because $\displaystyle a= 1/e^6$ and b= -6.