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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Blitzer 8th EditionCh. 4 - Exponential and Logarithmic Functions문제 27
5장, 문제 27

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

검증된 단계별 안내
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 \( \frac{1}{8} \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
2m
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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.
추천 영상:
7:30
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.
추천 영상:
5:36
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.
추천 영상:
5:14
Evaluate Logarithms
관련 실천
교과서 질문

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 5x=17

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교과서 질문

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. logb ((x2y)/z2)

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교과서 질문

Begin by graphing f(x) = 2x. Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. g(x) = 2x – 1

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교과서 질문

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log6 (36/(√(x+1))

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교과서 질문

Begin by graphing f(x) = 2x. Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. g(x) = 2x+1

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교과서 질문

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. ex=5.7

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