Problem

Fill in the missing values to make the equations true.
(a) $\log _{7} 8-\log _{7} 5=\log _{7}$
(b) $\log _{2} \square+\log _{2} 11=\log _{2} 55$
(c) $\log _{6} 9=\square \log _{6} 3$

Answer

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Answer

For (c), we use the property of logarithms that states \(\log_b{a^n} = n\log_b{a}\). So, we need to find the value that makes the log base 6 of 9 equal to the missing value times the log base 6 of 3. The solution is \(\boxed{2}\).

Steps

Step 1 :For (a), we use the property of logarithms that states \(\log_b{a} - \log_b{c} = \log_b{\frac{a}{c}}\). So, we need to find the value that makes \(\frac{8}{5}\) equal to the missing value. The solution is \(\boxed{1.6}\).

Step 2 :For (b), we use the property of logarithms that states \(\log_b{a} + \log_b{c} = \log_b{ac}\). So, we need to find the value that makes the product of the missing value and 11 equal to 55. The solution is \(\boxed{5}\).

Step 3 :For (c), we use the property of logarithms that states \(\log_b{a^n} = n\log_b{a}\). So, we need to find the value that makes the log base 6 of 9 equal to the missing value times the log base 6 of 3. The solution is \(\boxed{2}\).

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