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

Verified step by step guidance

1. Recall the logarithmic property that states: \(\ln a - \ln b = \ln \left( \frac{a}{b} \right)\) for positive values of \(a\) and \(b\).
2. Apply this property to the expression \(\ln(4x) - \ln(2x)\) by writing it as \(\ln \left( \frac{4x}{2x} \right)\).
3. Simplify the fraction inside the logarithm: \(\frac{4x}{2x} = 2\) since \(x > 0\) and cancels out.
4. Rewrite the expression as \(\ln(2)\) after simplification.
5. Compare the simplified expression \(\ln(2)\) with the given options to identify the equivalent expression.

Key Concepts

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.