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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblema 27
Capitolo 5, Problema 27

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

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

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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.
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Logarithms Introduction

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.
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Change of Base Property

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.
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Evaluate Logarithms
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