Find each logarithm without using a calculator or tables.
(a) $\ln \left(e^{16}\right)$
(b) $\ln (\sqrt{e})$
(c) $\ln \left(\sqrt[3]{e^{4}}\right)$
(d) $\ln (1)$
(e) $\ln \left(\ln \left(e^{e}\right)\right)$
(f) $\ln \left(\frac{1}{e^{4}}\right)$

Step 1 ：Given the property of logarithms, if $y = \ln(x)$, then $e^y = x$. We can use this property to simplify the expressions.

Step 2 ：For (a) $\ln \left(e^{16}\right)$, the exponent to which e must be raised to equal $e^{16}$ is 16. So, $\ln \left(e^{16}\right) = 16$

Step 3 ：For (b) $\ln (\sqrt{e})$, the exponent to which e must be raised to equal $\sqrt{e}$ is 0.5. So, $\ln (\sqrt{e}) = 0.5$

Step 4 ：For (c) $\ln \left(\sqrt[3]{e^{4}}\right)$, the exponent to which e must be raised to equal $\sqrt[3]{e^{4}}$ is approximately 1.3333333333333333. So, $\ln \left(\sqrt[3]{e^{4}}\right) \approx 1.3333333333333333$

Step 5 ：For (d) $\ln (1)$, the exponent to which e must be raised to equal 1 is 0. So, $\ln (1) = 0$

Step 6 ：For (e) $\ln \left(\ln \left(e^{e}\right)\right)$, the exponent to which e must be raised to equal $\ln \left(e^{e}\right)$ is 1. So, $\ln \left(\ln \left(e^{e}\right)\right) = 1$

Step 7 ：For (f) $\ln \left(\frac{1}{e^{4}}\right)$, the exponent to which e must be raised to equal $\frac{1}{e^{4}}$ is -4. So, $\ln \left(\frac{1}{e^{4}}\right) = -4$

Step 8 ：Final Answer: (a) \(\boxed{16}\), (b) \(\boxed{0.5}\), (c) \(\boxed{1.3333333333333333}\), (d) \(\boxed{0}\), (e) \(\boxed{1}\), (f) \(\boxed{-4}\)