Fill in the missing values to make the equations true. (a) $\log _{7} 3+\log _{7} 8=\log _{7} \square$ (b) $\log _{9} 10-\log _{9} \square=\log _{9} \frac{10}{7}$ (c) $\log _{9} \frac{1}{27}=\square \log _{9} 3$

Step 1 ：Given the equations are: (a) \(\log _{7} 3+\log _{7} 8=\log _{7} \square\), (b) \(\log _{9} 10-\log _{9} \square=\log _{9} \frac{10}{7}\), (c) \(\log _{9} \frac{1}{27}=\square \log _{9} 3\)

Step 2 ：We can use the logarithmic properties to solve these problems. The properties are as follows: 1. \(\log_b(mn) = \log_b(m) + \log_b(n)\), 2. \(\log_b(m/n) = \log_b(m) - \log_b(n)\), 3. \(\log_b(m^n) = n\log_b(m)\)

Step 3 ：For (a), we can use the first property to combine the two logarithms on the left side of the equation. The missing value will be the product of the numbers whose logarithms are being added. So, \(\log _{7} 3+\log _{7} 8=\log _{7} (3*8)\), hence the missing value is 24

Step 4 ：For (b), we can use the second property to separate the logarithm on the right side of the equation. The missing value will be the number that, when divided from 10, gives the number whose logarithm is on the right side of the equation. So, \(\log _{9} 10-\log _{9} \square=\log _{9} \frac{10}{7}\), hence the missing value is 7

Step 5 ：For (c), we can use the third property to separate the logarithm on the left side of the equation. The missing value will be the exponent that makes the number inside the logarithm equal to 1/27. So, \(\log _{9} \frac{1}{27}=\square \log _{9} 3\), hence the missing value is -3

Step 6 ：Final Answer: (a) \(\log _{7} 3+\log _{7} 8=\log _{7} \boxed{24}\), (b) \(\log _{9} 10-\log _{9} \boxed{7}=\log _{9} \frac{10}{7}\), (c) \(\log _{9} \frac{1}{27}=\boxed{-3} \log _{9} 3\)