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

Verified step by step guidance

1. Recall the logarithmic property that states: \(a \ln b = \ln b^a\). This means you can rewrite \$2 \ln(3x)\( as \)\ln((3x)^2)$.
2. Apply the exponent inside the logarithm: \((3x)^2 = 3^2 \times x^2 = 9x^2\). So, \$2 \ln(3x) = \ln(9x^2)$.
3. Recognize that \(\ln(9x^2)\) can be separated using the logarithm product rule: \(\ln(ab) = \ln a + \ln b\). Therefore, \(\ln(9x^2) = \ln 9 + \ln x^2\).
4. Use the power rule for logarithms on \(\ln x^2\): \(\ln x^2 = 2 \ln x\). So, \(\ln(9x^2) = \ln 9 + 2 \ln x\).
5. Compare the expression \(\ln(9x^2)\) with the given options to identify which one matches \$2 \ln(3x)$.

Key Concepts

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.