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 + 5 ln y - 6 ln z

검증된 단계별 안내

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 \(3 \ln x\) as \(\ln x^3\), \(5 \ln y\) as \(\ln y^5\), and \(-6 \ln z\) as \(\ln z^{-6}\).

Use the logarithmic property that the sum and difference of logarithms correspond to multiplication and division inside a single logarithm: \(\ln a + \ln b = \ln (a \cdot b)\) and \(\ln a - \ln b = \ln \left(\frac{a}{b}\right)\).

Combine the terms into a single logarithm: \(\ln \left( \frac{x^3 \cdot y^5}{z^6} \right)\).

Verify that the coefficient of the logarithm is 1, which it is, so the expression is now condensed into a single logarithm.

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Properties of Logarithms

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

Condensing Logarithmic Expressions

Condensing logarithmic expressions means rewriting multiple logs into a single logarithm. This is done by applying the product, quotient, and power rules to combine terms, ensuring the final expression has a coefficient of 1.

Evaluating Logarithms Without a Calculator

Evaluating logarithms without a calculator involves recognizing values of logs based on known bases and properties, such as log base e (natural logs) of 1 is 0, or using simplifications from the condensed form to find exact values.