Evaluate each expression without using a calculator. log₂ (1/√2)

Verified step by step guidance

1. Recall the logarithm property that allows you to rewrite the logarithm of a quotient or root: \(\log_b \left( \frac{1}{\sqrt{2}} \right) = \log_b (1) - \log_b (\sqrt{2})\).
2. Recognize that \(\sqrt{2}\) can be expressed as an exponent: \(\sqrt{2} = 2^{\frac{1}{2}}\).
3. Use the logarithm power rule: \(\log_b (a^c) = c \cdot \log_b (a)\), so \(\log_2 (\sqrt{2}) = \log_2 \left( 2^{\frac{1}{2}} \right) = \frac{1}{2} \cdot \log_2 (2)\).
4. Since \(\log_2 (2) = 1\), simplify the expression to \(\frac{1}{2} \cdot 1 = \frac{1}{2}\).
5. Recall that \(\log_2 (1) = 0\), so the original expression becomes \(0 - \frac{1}{2} = -\frac{1}{2}\).

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 helps in rewriting and evaluating logarithmic expressions.

Properties of Logarithms

Logarithms have key properties such as log_b(xy) = log_b(x) + log_b(y), log_b(x/y) = log_b(x) - log_b(y), and log_b(x^r) = r * log_b(x). These properties allow simplification of complex expressions without a calculator.

Exponents and Radicals

Radicals can be expressed as fractional exponents, e.g., √2 = 2^(1/2). Recognizing this allows rewriting expressions like 1/√2 as 2^(-1/2), which simplifies the evaluation of logarithms by converting roots into powers.