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\( \begin{array}{l} \text { Use properties of logarithms to expand each logarithmic expression as much as possible. Evaluate logarithmic expressions without using a calculator if possible. }\\ \log _{4}\left(\frac{4}{y}\right)\\ \log _{4}\left(\frac{4}{y}\right)=\square \end{array} \)

Ask by Chadwick Nichols. in the United States
Mar 21,2025

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Answer

\[ \log_4\left(\frac{4}{y}\right) = 1 - \log_4(y) \]

Solution

1. Write the expression using the quotient rule of logarithms: \[ \log_4\left(\frac{4}{y}\right) = \log_4(4) - \log_4(y) \] 2. Evaluate \(\log_4(4)\). Since the base and the argument are the same, we have: \[ \log_4(4) = 1 \] 3. Substitute back into the expression: \[ \log_4\left(\frac{4}{y}\right) = 1 - \log_4(y) \] Thus, the expanded form is: \[ 1 - \log_4(y) \]

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To expand \( \log_{4}\left(\frac{4}{y}\right) \), we can use the properties of logarithms. According to the quotient rule, we have: \[ \log_{b}\left(\frac{a}{c}\right) = \log_{b}(a) - \log_{b}(c) \] Applying this rule to our expression, we get: \[ \log_{4}\left(\frac{4}{y}\right) = \log_{4}(4) - \log_{4}(y) \] Since \( \log_{4}(4) = 1 \) (because \( 4^1 = 4 \)), our expression simplifies to: \[ \log_{4}\left(\frac{4}{y}\right) = 1 - \log_{4}(y) \] So, the final expanded result is: \[ \log_{4}\left(\frac{4}{y}\right) = 1 - \log_{4}(y) \]

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