Problem

Fill in the missing values to make the equations true.
(a) $\log _{2} 7-\log _{2} 5=\log _{2} \square$
(b) $\log _{5} 11+\log _{5} \square=\log _{5} 44$
(c) $-2 \log _{2} 3=\log _{2} \square$

Answer

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Answer

(c) \(\boxed{0.1111111111111111}\)

Steps

Step 1 :Use the logarithm properties to solve these equations. The properties are as follows:

Step 2 :1. \(\log_b{mn} = \log_b{m} + \log_b{n}\)

Step 3 :2. \(\log_b{\frac{m}{n}} = \log_b{m} - \log_b{n}\)

Step 4 :3. \(n\log_b{m} = \log_b{m^n}\)

Step 5 :For equation (a), use the second property to combine the two logarithms on the left side of the equation. The equation then becomes \(\log_2{\frac{7}{5}} = \log_2{\square}\). The value that makes this equation true is \(\frac{7}{5}\).

Step 6 :For equation (b), use the first property to combine the two logarithms on the left side of the equation. The equation then becomes \(\log_5{11 \times \square} = \log_5{44}\). The value that makes this equation true is \(\frac{44}{11}\).

Step 7 :For equation (c), use the third property to rewrite the left side of the equation. The equation then becomes \(\log_2{3^{-2}} = \log_2{\square}\). The value that makes this equation true is \(3^{-2}\).

Step 8 :Final Answer:

Step 9 :(a) \(\boxed{1.4}\)

Step 10 :(b) \(\boxed{4.0}\)

Step 11 :(c) \(\boxed{0.1111111111111111}\)

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