In Exercises 21–42, evaluate each expression without using a calculator. log5 (1/5)

Verified step by step guidance

1. Recognize that the expression \( \log_5 \left( \frac{1}{5} \right) \) is asking for the power to which 5 must be raised to get \( \frac{1}{5} \).
2. Recall the property of logarithms: \( \log_b \left( \frac{1}{b} \right) = -1 \). This is because \( b^{-1} = \frac{1}{b} \).
3. Apply this property to the given expression: \( \log_5 \left( \frac{1}{5} \right) = -1 \).
4. Understand that this is because \( 5^{-1} = \frac{1}{5} \), which confirms the logarithmic property used.
5. Conclude that the expression evaluates to \(-1\) based on the properties of logarithms.

Key Concepts

Logarithms

Logarithms are the inverse operations of exponentiation. The logarithm of a number is the exponent to which a base must be raised to produce that number. For example, if b^y = x, then log_b(x) = y. Understanding this relationship is crucial for evaluating logarithmic expressions.

Change of Base Formula

The change of base formula allows you to convert logarithms from one base to another. It states that log_b(a) can be expressed as log_k(a) / log_k(b) for any positive k. This is particularly useful when dealing with logarithms of bases that are not easily computable.

Properties of Logarithms

Logarithms have several key properties that simplify calculations. For instance, log_b(1) = 0 for any base b, since b^0 = 1. Additionally, log_b(b) = 1, as any number raised to the power of 1 is itself. These properties are essential for evaluating logarithmic expressions without a calculator.