In Exercises 21–42, evaluate each expression without using a calculator. log2 (1/8)

Verified step by step guidance

1. Recognize that the expression \( \log_2 \left( \frac{1}{8} \right) \) is asking for the power to which 2 must be raised to get \( \frac{1}{8} \).
2. Recall that \( 8 = 2^3 \), so \( \frac{1}{8} = \frac{1}{2^3} = 2^{-3} \).
3. Set the expression equal to \( x \): \( \log_2 \left( \frac{1}{8} \right) = x \).
4. Rewrite the equation in exponential form: \( 2^x = \frac{1}{8} \).
5. Since \( \frac{1}{8} = 2^{-3} \), equate the exponents: \( x = -3 \).

Key Concepts

Logarithms

Logarithms are the inverse operations of exponentiation. The logarithm log_b(a) answers the question: 'To what exponent must the base b be raised to produce a?' For example, log_2(8) = 3 because 2^3 = 8. Understanding logarithms is essential for evaluating expressions like log_2(1/8).

Change of Base Formula

The Change of Base Formula allows you to convert logarithms from one base to another, which can simplify calculations. It states that log_b(a) = log_k(a) / log_k(b) for any positive k. This is particularly useful when dealing with logarithms that are not easily computed in their original base.

Negative Exponents

Negative exponents represent the reciprocal of the base raised to the absolute value of the exponent. For instance, a^(-n) = 1/(a^n). In the context of logarithms, recognizing that 1/8 can be expressed as 2^(-3) helps in evaluating log_2(1/8) by transforming the expression into a more manageable form.