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. logb x^3

Key Concepts

Properties of Logarithms

The properties of logarithms are rules that simplify the manipulation of logarithmic expressions. Key properties include the product rule (logb(mn) = logb(m) + logb(n)), the quotient rule (logb(m/n) = logb(m) - logb(n)), and the power rule (logb(m^n) = n * logb(m)). Understanding these properties is essential for expanding and simplifying logarithmic expressions.

Power Rule of Logarithms

The power rule of logarithms states that the logarithm of a number raised to an exponent can be expressed as the exponent multiplied by the logarithm of the base number. For example, logb(x^3) can be rewritten as 3 * logb(x). This rule is particularly useful for expanding logarithmic expressions involving powers.

Logarithmic Evaluation

Evaluating logarithmic expressions involves finding the value of the logarithm based on known values or properties. For instance, if the base and the argument of the logarithm are known, one can directly compute the logarithm. In cases where the logarithm cannot be easily evaluated, properties of logarithms can help simplify the expression for easier computation.