Evaluate each expression without using a calculator. log₄ 16

Verified step by step guidance

1. Recognize that the expression \( \log_4 16 \) asks for the exponent to which the base 4 must be raised to get 16.
2. Express 16 as a power of 4. Since 16 is \( 4^2 \), rewrite the expression as \( \log_4 (4^2) \).
3. Use the logarithmic identity \( \log_b (b^k) = k \) to simplify \( \log_4 (4^2) \) to 2.
4. Conclude that \( \log_4 16 = 2 \) because 4 raised to the power 2 equals 16.
5. Thus, the value of the logarithm is the exponent 2.

Key Concepts

Logarithm Definition

A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log_b(a) = c means b^c = a. Understanding this definition is essential to evaluate logarithmic expressions.

Properties of Exponents

Since logarithms are closely related to exponents, knowing how to express numbers as powers of a base helps simplify logarithms. For instance, recognizing that 16 = 4^2 allows you to rewrite log_4(16) as log_4(4^2).

Logarithm Power Rule

The power rule states that log_b(a^n) = n * log_b(a). This property allows you to bring the exponent in the argument down as a multiplier, simplifying the evaluation of logarithms when the argument is a power of the base.