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 9x^2

Key Concepts

Properties of Logarithms

Logarithms have specific properties that simplify expressions. Key properties include the product rule (ln(a) + ln(b) = ln(ab)), the quotient rule (ln(a) - ln(b) = ln(a/b)), and the power rule (k * ln(a) = ln(a^k)). Understanding these properties is essential for manipulating logarithmic expressions effectively.

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 derivative of ln(x) being 1/x. Recognizing how to work with ln is crucial for solving logarithmic equations.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a * b^x, where a is a constant, b is the base, and x is the exponent. The relationship between logarithms and exponential functions is fundamental, as logarithms can be used to solve for exponents. This concept is vital for understanding how to manipulate and solve logarithmic expressions.