Evaluate the logarithmic expression without using a calculator. Remember that $\log _{a} x$ is the exponent to which a must be raised in order to obtain $\mathrm{x}$. (a) $\log _{3} 81$ (d) $\log _{6} \sqrt{6}$ (b) $\log _{3} \frac{1}{3}$ (e) $\log _{e} 1$ (c) $\log _{10} 0.01$ (f) $\log _{3} 27^{3 / 2}$

Step 1 ：Let's start with the first question (a) $\log _{3} 81$. Let $x=\log_3 81$. Then, we must have $3^x = 81 = 3^4$, so $x=\boxed{4}$.

Step 2 ：For the second question (d) $\log _{6} \sqrt{6}$, let $x=\log_6 \sqrt{6}$. Then, we must have $6^x = \sqrt{6} = 6^{\frac{1}{2}}$, so $x=\boxed{\frac{1}{2}}$.

Step 3 ：For the third question (b) $\log _{3} \frac{1}{3}$, let $x=\log_3\frac{1}{3}$. Then, we must have $3^x = \frac{1}{3} = 3^{-1}$, so $x=\boxed{-1}$.

Step 4 ：For the fourth question (e) $\log _{e} 1$, let $x=\log_e 1$. Then, we must have $e^x = 1 = e^0$, so $x=\boxed{0}$.

Step 5 ：For the fifth question (c) $\log _{10} 0.01$, let $x=\log_{10} 0.01$. Then, we must have $10^x = 0.01 = 10^{-2}$, so $x=\boxed{-2}$.

Step 6 ：For the sixth question (f) $\log _{3} 27^{3 / 2}$, let $x=\log_3 27^{3 / 2}$. Then, we must have $3^x = 27^{3 / 2} = 3^9$, so $x=\boxed{9}$.