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. 5 ln x - 2 ln y

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

Apply this property to each term: rewrite \(5 \ln x\) as \(\ln x^5\) and \(-2 \ln y\) as \(\ln y^{-2}\).

Use the logarithmic property that the difference of logarithms is the logarithm of a quotient: \(\ln a - \ln b = \ln \left( \frac{a}{b} \right)\).

Combine the two logarithms into a single logarithm: \(\ln x^5 - \ln y^2 = \ln \left( \frac{x^5}{y^2} \right)\).

Write the final condensed expression as a single logarithm with coefficient 1: \(\ln \left( \frac{x^5}{y^2} \right)\).

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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, i.e., a·ln(b) = ln(b^a).

Condensing Logarithmic Expressions

Condensing logarithmic expressions means rewriting multiple logarithms as a single logarithm. This is done by applying the product rule (log a + log b = log(ab)), quotient rule (log a - log b = log(a/b)), and power rule to combine terms into one logarithm with coefficient 1.

Evaluating Logarithms Without a Calculator

Evaluating logarithms without a calculator involves recognizing values that simplify to known logarithmic results, such as ln(e) = 1 or ln(1) = 0. Simplifying expressions using properties can sometimes reduce the argument to these known values, allowing exact evaluation.