Use properties of logarithms to expand the logarithmic expression below as much as possible.
\[
\log _{d} \frac{\sqrt{a} b^{7}}{c^{4}}
\]
\[
\log _{d} \frac{\sqrt{a} b^{7}}{c^{4}}=\square \text { (Simplify your answer.) }
\]

Step 1 ：Given the expression \(\log _{d} \frac{\sqrt{a} b^{7}}{c^{4}}\)

Step 2 ：We can use the properties of logarithms to simplify this expression.

Step 3 ：First, we use the property \(\log_b(m/n) = \log_b(m) - \log_b(n)\) to split the fraction inside the logarithm: \(\log _{d} \sqrt{a} b^{7} - \log _{d} c^{4}\)

Step 4 ：Next, we use the property \(\log_b(mn) = \log_b(m) + \log_b(n)\) to split the multiplication inside the first logarithm: \(\log _{d} \sqrt{a} + \log _{d} b^{7} - \log _{d} c^{4}\)

Step 5 ：Then, we use the property \(\log_b(m^n) = n \log_b(m)\) to bring down the exponents: \(\frac{1}{2} \log _{d} a + 7 \log _{d} b - 4 \log _{d} c\)

Step 6 ：Finally, we can rewrite the expression in terms of natural logarithms to get the final answer: \(\boxed{\frac{\log a}{2 \log d}+7 \frac{\log b}{\log d}-4 \frac{\log c}{\log d}}\)