Evaluate each expression without using a calculator. log₅ 5

Verified step by step guidance

1. Recall the definition of a logarithm: \(\log_b a = c\) means that \(b^c = a\).
2. In this problem, we have \(\log_5 5\), which asks: "To what power must 5 be raised to get 5?"
3. Since \$5^1 = 5$, the exponent that satisfies this equation is 1.
4. Therefore, \(\log_5 5 = 1\) because the base and the argument are the same.
5. This is a general property of logarithms: \(\log_b b = 1\) for any positive base \(b \neq 1\).

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 a Base to Itself

The logarithm of a base raised to itself, such as log_b(b), always equals 1 because the base raised to the power 1 equals itself. This property simplifies expressions like log5 5 directly to 1.

Properties of Logarithms

Logarithms follow specific properties, such as log_b(b^x) = x and log_b(1) = 0. Recognizing these properties helps in simplifying and evaluating logarithmic expressions without a calculator.