Evaluate each expression without using a calculator. log₄ 1

Verified step by step guidance

1. Recall the definition of logarithm: \(\log_b a = c\) means that \(b^c = a\).
2. In this problem, we have \(\log_4 1\), so we want to find the exponent \(c\) such that \$4^c = 1$.
3. Remember that any nonzero number raised to the power of 0 equals 1, i.e., \(b^0 = 1\) for \(b \neq 0\).
4. Since \$4^0 = 1\(, it follows that \)\log_4 1 = 0$.
5. Therefore, the value of \(\log_4 1\) is the exponent that makes the base 4 equal to 1, which is 0.

Key Concepts

Definition of Logarithms

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.

Logarithm of 1

For any positive base b (b ≠ 1), log_b(1) is always 0 because b raised to the power 0 equals 1. This property simplifies evaluating logarithms where the argument is 1.

Properties of Logarithms Without a Calculator

Evaluating logarithms without a calculator relies on recognizing special values and applying logarithmic properties, such as log_b(b) = 1 and log_b(1) = 0. Familiarity with these properties allows quick simplification.