Evaluate each expression without using a calculator. log₅ 5⁷

Verified step by step guidance

1. Recognize that the expression is \( \log_5 5^7 \), which is a logarithm with base 5 of \( 5^7 \).
2. Recall the logarithmic identity: \( \log_b b^x = x \), which means the logarithm of a base raised to a power is just the exponent.
3. Apply this identity directly to the expression: \( \log_5 5^7 = 7 \).
4. Therefore, the value of \( \log_5 5^7 \) simplifies to the exponent 7 without further calculation.
5. This shows how logarithms and exponents are inverse operations, making such expressions straightforward to evaluate.

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 for evaluating logarithmic expressions.

Logarithm of a Power

The logarithm of a number raised to an exponent can be simplified using the rule log_b(a^c) = c * log_b(a). This property allows you to bring the exponent in front as a multiplier, simplifying the evaluation process.

Logarithm of the Base

When the argument of a logarithm is the same as its base, log_b(b) equals 1 because b^1 = b. This fact helps simplify expressions like log5(5^7) by reducing the inner logarithm to a simple exponent.