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. $$\frac{1}{3}$ $\left$[ 2 $\ln$(x + 5) - $\ln$ x - $\ln$ (x^2 - 4) $\right$]$

검증된 단계별 안내

Start with the given expression: $\frac{1}{3} \left[ 2 \ln(x + 5) - \ln x - \ln(x^2 - 4) \right]$.

Use the logarithm power rule to move coefficients inside the logarithms as exponents: $2 \ln(x + 5) = \ln((x + 5)^2)$.

Rewrite the expression inside the brackets as a single logarithm using the properties of logarithms: $\ln((x + 5)^2) - \ln x - \ln(x^2 - 4) = \ln \left( \frac{(x + 5)^2}{x (x^2 - 4)} \right)$.

Recall that $x^2 - 4$ is a difference of squares and can be factored as $(x - 2)(x + 2)$, so rewrite the denominator accordingly: $x (x - 2)(x + 2)$.

Now apply the outer coefficient $\frac{1}{3}$ as a power to the entire logarithm: $\frac{1}{3} \ln \left( \frac{(x + 5)^2}{x (x - 2)(x + 2)} \right) = \ln \left( \left( \frac{(x + 5)^2}{x (x - 2)(x + 2)} \right)^{\frac{1}{3}} \right)$.

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주요 개념

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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, log(a) + log(b) = log(ab), log(a) - log(b) = log(a/b), and k·log(a) = log(a^k). Understanding these is essential to condense multiple logarithms into a single expression.

Natural Logarithm (ln)

The natural logarithm, denoted ln, is the logarithm with base e (approximately 2.718). It has the same properties as other logarithms but is commonly used in calculus and algebra. Recognizing ln and its behavior helps in simplifying expressions and evaluating them when possible.

Simplifying Algebraic Expressions Inside Logarithms

Before condensing logarithms, simplify the algebraic expressions inside them, such as factoring or recognizing differences of squares (e.g., x^2 - 4 = (x-2)(x+2)). This simplification aids in combining logarithms correctly and can make evaluation or further manipulation easier.