How do you simplify ln (1/e^3)?

$\ln \left(\frac{1}{e} ^ \left(3\right)\right) = \ln \left({e}^{- 3}\right) = - 3$.
Since $\ln \left(x\right)$ and ${e}^{x}$ are inverse functions, $\ln \left({e}^{x}\right) = x$ for all values of $x$.
Since $\frac{1}{e} ^ \left\{3\right\} = {e}^{- 3}$ by definition of negative exponents, it follows that $\ln \left(\frac{1}{e} ^ \left(3\right)\right) = \ln \left({e}^{- 3}\right) = - 3$.
You could also note that $\ln \left(\frac{1}{e} ^ \left\{3\right\}\right) = \ln \left(1\right) - \ln \left({e}^{3}\right) = 0 - 3 = - 3$ since $\ln \left(\frac{A}{B}\right) = \ln \left(A\right) - \ln \left(B\right)$ and $\ln \left(1\right) = 0$.