Work each problem. Which of the following is equivalent to ln(4x) - ln(2x) for x \u003e 0? A. 2 ln x B. ln 2x C. (ln 4x)/(ln 2x) D. ln 2

Ch. 4 - Inverse, Exponential, and Logarithmic Functions

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Lial 13th Edition

Ch. 4 - Inverse, Exponential, and Logarithmic Functions

문제 100

5장, 문제 100

Work each problem. Which of the following is equivalent to ln(4x) - ln(2x) for x > 0? A. 2 ln x B. ln 2x C. (ln 4x)/(ln 2x) D. ln 2

검증된 단계별 안내

Recall the logarithmic property that states: \(\ln a - \ln b = \ln \left( \frac{a}{b} \right)\) for positive values of \(a\) and \(b\).

Apply this property to the expression \(\ln(4x) - \ln(2x)\) by writing it as \(\ln \left( \frac{4x}{2x} \right)\).

Simplify the fraction inside the logarithm: \(\frac{4x}{2x} = 2\) since \(x > 0\) and cancels out.

Rewrite the expression as \(\ln(2)\) after simplification.

Compare the simplified expression \(\ln(2)\) with the given options to identify the equivalent expression.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Properties of Logarithms

Logarithms have specific properties that simplify expressions, such as the difference rule: ln(a) - ln(b) = ln(a/b). This property allows combining or breaking down logarithmic expressions by converting subtraction into division inside a single logarithm.

Domain Restrictions for Logarithmic Functions

The argument of a logarithm must be positive, so for ln(4x) and ln(2x), x must be greater than zero. Understanding domain restrictions ensures the expression is valid and helps avoid undefined values during simplification.

Simplifying Algebraic Expressions Inside Logarithms

When simplifying ln(4x) - ln(2x), use algebraic manipulation inside the logarithm: (4x)/(2x) simplifies to 2, since x > 0. Recognizing how to simplify fractions inside logarithms is key to finding the equivalent expression.