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. 3 ln x - (1/3) ln y

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Recall the logarithmic property that allows you to move coefficients in front of logarithms as exponents inside the log: \(a \ln b = \ln b^{a}\).

Apply this property to each term: rewrite \(3 \ln x\) as \(\ln x^{3}\) and \(\frac{1}{3} \ln y\) as \(\ln y^{\frac{1}{3}}\).

Rewrite the original expression using these new forms: \(\ln x^{3} - \ln y^{\frac{1}{3}}\).

Use the logarithmic subtraction property: \(\ln a - \ln b = \ln \left( \frac{a}{b} \right)\) to combine the two logarithms into one.

Write the final condensed expression as a single logarithm: \(\ln \left( \frac{x^{3}}{y^{\frac{1}{3}}} \right)\).

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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.

Condensing Logarithmic Expressions

Condensing logarithmic expressions means rewriting multiple logarithms as a single logarithm. This involves applying the properties of logarithms to combine terms, such as turning sums into products and differences into quotients inside the log. The goal is to have one logarithm with a coefficient of 1.

Evaluating Logarithms Without a Calculator

Evaluating logarithms without a calculator involves recognizing logarithmic values based on known log properties or special values, such as log base 10 of 1 is 0, or log base e of e is 1. Simplifying expressions using these known values helps in exact evaluation when possible.