In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. ln(e^2/5)

Key Concepts

Properties of Logarithms

The properties of logarithms are rules that simplify the manipulation of logarithmic expressions. Key properties include the product rule (log(a*b) = log(a) + log(b)), the quotient rule (log(a/b) = log(a) - log(b)), and the power rule (log(a^b) = b*log(a)). Understanding these properties is essential for expanding and simplifying 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 exponential growth models. Recognizing that ln(e^x) simplifies to x is crucial for evaluating expressions involving natural logarithms.

Exponentiation and Logarithmic Relationships

Exponentiation and logarithmic functions are inverses of each other. This means that if y = e^x, then x = ln(y). This relationship allows for the simplification of expressions involving exponents and logarithms, making it easier to evaluate and expand logarithmic expressions like ln(e^2/5) by breaking them down into manageable parts.