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. (1/2)ln x + ln y

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Recall the logarithmic property that allows you to move a coefficient in front of a logarithm as an exponent inside the logarithm: \(a \ln b = \ln b^{a}\).

Apply this property to the first term: \((1/2) \ln x = \ln x^{1/2}\).

Rewrite the expression using this result: \(\ln x^{1/2} + \ln y\).

Use the logarithmic property that the sum of logarithms is the logarithm of the product: \(\ln a + \ln b = \ln (a \cdot b)\).

Combine the terms into a single logarithm: \(\ln \left(x^{1/2} \cdot y\right)\).

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Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. These allow combining or breaking down logarithmic expressions. For example, the power rule states that a coefficient in front of a logarithm can be rewritten as an exponent inside the log, which is essential for condensing expressions.

Power Rule of Logarithms

The power rule states that a coefficient multiplied by a logarithm can be rewritten as the logarithm of the argument raised to that coefficient's power: a * ln(b) = ln(b^a). This is crucial for rewriting (1/2)ln x as ln(x^(1/2)) to combine terms into a single logarithm.

Combining Logarithms Using the Product Rule

The product rule states that the sum of two logarithms with the same base can be combined into a single logarithm of the product of their arguments: ln(a) + ln(b) = ln(ab). This allows rewriting ln(x^(1/2)) + ln(y) as ln(x^(1/2) * y), condensing the expression into one logarithm.