Fill in the missing values to make the equations true. (a) $\log _{5} 9-\log _{5} 11=\log _{5}$ [] (b) $\log _{9} 10+\log _{9} \square=\log _{9} 70$ (c) $\log _{7} \frac{1}{16}=\square \log _{7} 2$
Solution
Step 1 ：First, we need to understand the properties of logarithms. The subtraction of two logarithms with the same base is equivalent to the logarithm of the division of their arguments. Similarly, the addition of two logarithms with the same base is equivalent to the logarithm of the multiplication of their arguments. Lastly, the logarithm of a number raised to an exponent is equivalent to the product of the exponent and the logarithm of the number.
Step 2 ：For equation (a), we can use the property of subtraction of logarithms. So, we divide 9 by 11 to get the missing value. The missing value is \( \boxed{0.8181818181818182} \).
Step 3 ：For equation (b), we can use the property of addition of logarithms. So, we divide 70 by 10 to get the missing value. The missing value is \( \boxed{7.0} \).
Step 4 ：For equation (c), we can use the property of logarithms of a number raised to an exponent. We know that \( \frac{1}{16} \) is equivalent to \( 2^{-4} \). So, the missing value is \( \boxed{4.0} \).
