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. 4 ln (x + 6) - 3 ln x

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Identify the logarithmic expression given: \(4 \ln (x + 6) - 3 \ln x\).

Use the power rule of logarithms, which states that \(a \ln b = \ln (b^a)\), to rewrite each term: \(4 \ln (x + 6) = \ln ((x + 6)^4)\) and \(3 \ln x = \ln (x^3)\).

Rewrite the expression using these powers: \(\ln ((x + 6)^4) - \ln (x^3)\).

Apply the logarithmic subtraction rule, which states \(\ln A - \ln B = \ln \left( \frac{A}{B} \right)\), to combine the terms into a single logarithm: \(\ln \left( \frac{(x + 6)^4}{x^3} \right)\).

The expression is now condensed into a single logarithm with coefficient 1: \(\ln \left( \frac{(x + 6)^4}{x^3} \right)\). No further simplification is possible without specific values for \(x\).

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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 lets you move coefficients as exponents inside the log, which is essential for condensing expressions.

Power Rule of Logarithms

The power rule states that a coefficient in front of a logarithm can be rewritten as an exponent inside the log: a * ln(b) = ln(b^a). This is crucial for rewriting expressions like 4 ln(x + 6) as ln((x + 6)^4) to combine terms into a single logarithm.

Combining Logarithmic Expressions

Using the product and quotient rules, multiple logarithms can be combined into one. The product rule states ln(a) + ln(b) = ln(ab), and the quotient rule states ln(a) - ln(b) = ln(a/b). Applying these helps write the expression as a single logarithm with coefficient 1.