교과서 질문
Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 42x−1=64
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Ch. 4 - Exponential and Logarithmic Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
5장, 문제 7
Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log7 (7/x)
검증된 단계별 안내1
Recall the logarithmic property that states \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \). This allows us to rewrite the logarithm of a quotient as the difference of two logarithms.
Apply this property to the given expression \( \log_7 \left( \frac{7}{x} \right) \), which becomes \( \log_7 7 - \log_7 x \).
Recognize that \( \log_7 7 \) asks the question: "To what power must 7 be raised to get 7?" Since 7 to the power of 1 is 7, \( \log_7 7 = 1 \).
Substitute this value back into the expression to get \( 1 - \log_7 x \).
The expression is now fully expanded using logarithmic properties and simplified as much as possible without a calculator.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Properties of Logarithms
Properties of logarithms include rules such as the product, quotient, and power rules. These allow us to rewrite logarithmic expressions in simpler or expanded forms. For example, the quotient rule states that log_b(M/N) = log_b(M) - log_b(N), which is essential for expanding expressions like log_7(7/x).
추천 영상:
5:36
Change of Base Property
Logarithm of the Base
The logarithm of a number to its own base is always 1, i.e., log_b(b) = 1. This fact helps simplify expressions where the argument of the logarithm is the base itself, such as log_7(7), which equals 1. Recognizing this can simplify parts of the expression without a calculator.
추천 영상:
7:30Logarithms Introduction
Evaluating Logarithms Without a Calculator
Evaluating logarithms without a calculator involves using known values and properties to simplify expressions. For example, recognizing that log_7(7) = 1 and applying the quotient rule can reduce the expression to a simpler form. This skill is important for exact answers in algebraic contexts.
추천 영상:
5:14Evaluate Logarithms
관련 실천
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교과서 질문
Write each equation in its equivalent exponential form. log6 216 = y
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