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Chapter 3: Problem 52

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$\frac{1}{3} \ln x+\ln y$$

Short Answer

Expert verified
The expression simplifies into a single logarithm which is \(\ln (x^{1/3}y)\).

Step by step solution

01

Recognize and Apply the Power Rule

The power rule states that \(a \ln b = \ln (b^a)\). So apply this rule to \(\frac{1}{3} \ln x\) to remove the coefficient 'a', i.e., \(1/3\). This gives us \(\ln x^{1/3}\) and the expression becomes \(\ln x^{1/3} + \ln y\).
02

Use the Product Rule

The product rule states that \(\ln a + \ln b = \ln (a \cdot b)\). So apply this rule to \(\ln x^{1/3} + \ln y\). This gives us the expression as \(\ln (x^{1/3} \cdot y)\).
03

Simplify the Expression

The logarithm of a product of terms is equivalent to the product of logarithms of individual terms. So simplify the expression \(\ln (x^{1/3} \cdot y)\) to the single logarithmic form \(\ln (x^{1/3}y)\).

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