In Exercises 21–42, evaluate each expression without using a calculator. log3 27

Key Concepts

Logarithms

A logarithm is the inverse operation to exponentiation, representing the power to which a base must be raised to obtain a given number. For example, in the expression log_b(a) = c, b^c = a. Understanding logarithms is essential for solving problems involving exponential growth or decay, as well as for simplifying complex expressions.

Change of Base Formula

The change of base formula allows you to convert logarithms from one base to another, which is particularly useful when the base is not easily computable. The formula states that log_b(a) = log_k(a) / log_k(b) for any positive k. This concept is crucial when evaluating logarithmic expressions without a calculator, as it enables the use of more familiar bases like 10 or e.

Exponential Relationships

Exponential relationships describe how a quantity grows or decays at a constant rate, represented mathematically as y = b^x, where b is the base and x is the exponent. Understanding these relationships helps in interpreting logarithmic expressions, as logarithms can be used to solve for the exponent in an exponential equation, providing insight into the behavior of exponential functions.