Use properties of logarithms to expand the logarithmic expression as much as possible. Evaluate logarithmic expressions without using a calculator if possible.
\[
\log _{4} \sqrt[7]{\frac{s^{8} t}{16}}
\]
\[
\log _{4} \sqrt[7]{\frac{s^{8} t}{16}}=
\]
(Use integers or fractions for any numbers in the expression.)

Step 1 ：The given expression is a logarithm with base 4. The argument of the logarithm is a seventh root of a fraction. We can use the properties of logarithms to simplify this expression.

Step 2 ：The properties of logarithms that we will use are: \(\log_b{a^n} = n \log_b{a}\) - the power rule, which allows us to bring an exponent out in front of the logarithm, \(\log_b{\frac{a}{c}} = \log_b{a} - \log_b{c}\) - the quotient rule, which allows us to turn a division inside a logarithm into a subtraction of two logarithms, \(\log_b{\sqrt[n]{a}} = \frac{1}{n} \log_b{a}\) - the root rule, which allows us to turn a root inside a logarithm into a division of the logarithm by the root's index.

Step 3 ：We can apply these rules in the following order: Apply the root rule to bring the seventh root outside of the logarithm, Apply the quotient rule to separate the numerator and denominator of the fraction inside the logarithm, Apply the power rule to bring the exponent of \(s\) outside of the logarithm.

Step 4 ：The expanded expression is \(\frac{1}{14} \log _{2}\left(s^{8} t\right)-\frac{2}{7}\). However, we can further simplify this expression by applying the power rule to the term \(\log _{2}\left(s^{8} t\right)\).

Step 5 ：The further expanded expression is \(\frac{4}{7} \log _{2}(s)+\frac{1}{14} \log _{2}(t)-\frac{2}{7}\). This is the most simplified form of the given logarithmic expression.

Step 6 ：Final Answer: \(\boxed{\frac{4}{7} \log _{2}(s)+\frac{1}{14} \log _{2}(t)-\frac{2}{7}}\)