Evaluate each expression without using a calculator. log₂ (1/8)

Verified step by step guidance

1. Recognize that the expression is \( \log_2 \left( \frac{1}{8} \right) \), which asks for the exponent to which 2 must be raised to get \( \frac{1}{8} \).
2. Rewrite \( \frac{1}{8} \) as a power of 2. Since \( 8 = 2^3 \), then \( \frac{1}{8} = 2^{-3} \).
3. Substitute this back into the logarithm: \( \log_2 \left( 2^{-3} \right) \).
4. Use the logarithmic identity \( \log_b (b^x) = x \) to simplify the expression to \( -3 \).
5. Conclude that \( \log_2 \left( \frac{1}{8} \right) = -3 \), meaning 2 raised to the power of -3 equals \( \frac{1}{8} \).

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 base 2 of 8 asks, '2 raised to what power equals 8?' Understanding this definition is essential to evaluate logarithmic expressions.

Properties of Exponents and Logarithms

Logarithms and exponents are inverse operations. Knowing that 1/8 can be written as 2 to the power of -3 (since 8 = 2^3) allows rewriting the logarithm in terms of exponents, making it easier to evaluate without a calculator.

Evaluating Logarithms of Fractions

When evaluating logarithms of fractions, express the fraction as a power of the base with a negative exponent. For example, log2(1/8) becomes log2(2^-3), which simplifies to -3 by the logarithm definition.