Question
Use the properties of logarithms to rewrite the following express \[ 2 \log _{4} x-9 \log _{4} y \] \( 2 \log _{4} x-9 \log _{4} y=\square \)
Ask by Luna Reeves. in the United States
Nov 07,2024
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Answer
\(2 \log_{4} x - 9 \log_{4} y = \log_{4} \left(\frac{x^{2}}{y^{9}}\right)\)
Solution
To rewrite the expression \(2 \log_{4} x - 9 \log_{4} y\) using the properties of logarithms, we can use the following properties:
1. \(a \log_{b} x = \log_{b} x^{a}\)
2. \(\log_{b} x - \log_{b} y = \log_{b} \left(\frac{x}{y}\right)\)
Applying these properties, we can rewrite the expression as follows:
\[2 \log_{4} x - 9 \log_{4} y = \log_{4} x^{2} - \log_{4} y^{9} = \log_{4} \left(\frac{x^{2}}{y^{9}}\right)\]
Therefore, the expression \(2 \log_{4} x - 9 \log_{4} y\) can be rewritten as \(\log_{4} \left(\frac{x^{2}}{y^{9}}\right)\).
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Using the properties of logarithms, we can rewrite \( 2 \log_{4} x \) as \( \log_{4} x^2 \) and \( 9 \log_{4} y \) as \( \log_{4} y^9 \). Therefore, we can express the original expression as: \[ 2 \log_{4} x - 9 \log_{4} y = \log_{4} x^2 - \log_{4} y^9 \] Now, applying the property of logarithms that states \( \log_{a} b - \log_{a} c = \log_{a} \left( \frac{b}{c} \right) \), we can further simplify it to: \[ \log_{4} \left( \frac{x^2}{y^9} \right) \] So, \(\boxed{\log_{4} \left( \frac{x^2}{y^9} \right)}\).
