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Chapter 13: Problem 40

Write as the sum or difference of logarithms and simplify, if possible. Assume all variables represent positive real numbers. $$\log _{3} \frac{\sqrt{x}}{y^{4}}$$

Short Answer

Expert verified
The simplified expression is \( \frac{1}{2}\log_{3}x - 4\log_{3}y \).

Step by step solution

01

Simplify the argument inside the logarithm

The argument is given as \(\frac{\sqrt{x}}{y^{4}}\). We can rewrite the square root, \(\sqrt{x}\), as a power of \(x\): \(x^\frac{1}{2}\). So the expression becomes \(\log_{3} \frac{x^{\frac{1}{2}}}{y^{4}}\).
02

Use the logarithm property of division

The logarithm property of division states that \[\log_{a} \frac{x}{y} = \log_{a} x - \log_{a} y\] Using this property, we can rewrite our expression as: \[\log_{3} \frac{x^{\frac{1}{2}}}{y^{4}} = \log_{3}(x^{\frac{1}{2}}) - \log_{3}(y^{4})\]
03

Use the logarithm property of exponentiation

The logarithm property of exponentiation states that \[\log_{a} x^n = n\log_{a}x\] Applying this property to each term, we get: \[\log_{3}(x^{\frac{1}{2}}) - \log_{3}(y^{4}) = \frac{1}{2}\log_{3}x - 4\log_{3}y\] So, the final expression simplified is: \[\frac{1}{2}\log_{3}x - 4\log_{3}y\]

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