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

Key Concepts

Properties of Logarithms

Logarithms have specific properties that simplify expressions. One key property is the quotient rule, which states that ln(a) - ln(b) = ln(a/b). This property allows us to combine logarithmic expressions by converting subtraction into division, making it essential for solving problems involving logarithms.

Natural Logarithm (ln)

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. It is commonly used in calculus and algebra due to its unique properties, such as the fact that the derivative of ln(x) is 1/x. Understanding how to manipulate ln expressions is crucial for solving logarithmic equations.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions involves applying logarithmic properties to rewrite them in a more manageable form. This includes using the product rule (ln(ab) = ln(a) + ln(b)) and the quotient rule. Mastery of these simplification techniques is vital for solving problems that require finding equivalent expressions.