Evaluate each expression without using a calculator. log₂ 64

Verified step by step guidance

1. Recognize that the expression \( \log_2 64 \) asks for the exponent to which the base 2 must be raised to get 64.
2. Express 64 as a power of 2. Since \( 64 = 2^6 \), rewrite the expression as \( \log_2 (2^6) \).
3. Use the logarithmic identity \( \log_b (b^x) = x \) to simplify \( \log_2 (2^6) \) to 6.
4. Therefore, \( \log_2 64 = 6 \), meaning 2 raised to the 6th power equals 64.
5. This completes the evaluation without using a calculator.

Key Concepts

Logarithms and Their 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 64 asks, '2 raised to what power equals 64?' Understanding this definition is essential to evaluate logarithmic expressions.

Properties of Exponents

Since logarithms are the inverse of exponents, knowing how to express numbers as powers of a base helps in evaluating logs. For instance, recognizing that 64 = 2^6 allows you to rewrite the logarithm in terms of exponents and solve it easily.

Evaluating Logarithms Without a Calculator

To evaluate logarithms without a calculator, rewrite the argument as a power of the base and then use the definition of logarithms. This method relies on mental math and familiarity with common powers, enabling quick and accurate solutions.