Work each problem. Which of the following is equivalent to 2 ln(3x) for x \u003e 0? A. ln 9 + ln x B. ln 6x C. ln 6 + ln x D. ln 9x2

Ch. 4 - Inverse, Exponential, and Logarithmic Functions

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Lial 13th Edition

Ch. 4 - Inverse, Exponential, and Logarithmic Functions

문제 99

5장, 문제 99

Work each problem. Which of the following is equivalent to 2 ln(3x) for x > 0?
A. ln 9 + ln x
B. ln 6x
C. ln 6 + ln x
D. ln 9x²

검증된 단계별 안내

Recall the logarithmic property that states: \(a \ln b = \ln b^a\). This means you can rewrite \(2 \ln(3x)\) as \(\ln((3x)^2)\).

Apply the exponent inside the logarithm: \((3x)^2 = 3^2 \times x^2 = 9x^2\). So, \(2 \ln(3x) = \ln(9x^2)\).

Recognize that \(\ln(9x^2)\) can be separated using the logarithm product rule: \(\ln(ab) = \ln a + \ln b\). Therefore, \(\ln(9x^2) = \ln 9 + \ln x^2\).

Use the power rule for logarithms on \(\ln x^2\): \(\ln x^2 = 2 \ln x\). So, \(\ln(9x^2) = \ln 9 + 2 \ln x\).

Compare the expression \(\ln(9x^2)\) with the given options to identify which one matches \(2 \ln(3x)\).

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주요 개념

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Properties of Logarithms

Logarithms have specific properties that simplify expressions, such as the product rule (ln a + ln b = ln(ab)) and the power rule (k ln a = ln(a^k)). These rules allow rewriting complex logarithmic expressions into simpler or equivalent forms.

Power Rule for Logarithms

The power rule states that multiplying a logarithm by a constant is equivalent to taking the logarithm of the argument raised to that constant: k ln(a) = ln(a^k). This is essential for rewriting expressions like 2 ln(3x) as ln((3x)^2).

Domain Restrictions in Logarithmic Functions

Logarithmic functions are defined only for positive arguments. For ln(3x), the domain restriction x > 0 ensures the argument 3x is positive, which is necessary for the expression to be valid and for applying logarithmic properties correctly.

Work each problem. Which of the following is equivalent to 2 ln(3x) for x > 0? A. ln 9 + ln x B. ln 6x C. ln 6 + ln x D. ln 9x2