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 5√x (fifth root of)

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 logarithmic expressions effectively.

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. It is commonly used in mathematics, particularly in calculus and exponential growth models. Recognizing that ln(x) can be expressed in terms of other logarithmic properties is crucial for solving logarithmic problems.

Radicals, such as the fifth root, can be expressed as exponents. For example, the fifth root of x can be written as x^(1/5). This understanding allows for the application of the power rule in logarithms, enabling the expansion of logarithmic expressions involving roots and powers.