Domanda del libro di testo
Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. log3 (7) = 1/[log7 (3)]
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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
Capitolo 5, Problema 105
In Exercises 105–108, evaluate each expression without using a calculator. log5 (log7 7)
Guida verificata passo dopo passo1
Recognize that the expression is \( \log_5 (\log_7 7) \), which means the logarithm base 5 of the logarithm base 7 of 7.
Evaluate the inner logarithm first: \( \log_7 7 \). Recall that \( \log_b b = 1 \) for any base \( b > 0 \) and \( b \neq 1 \).
Since \( \log_7 7 = 1 \), substitute this value back into the original expression to get \( \log_5 1 \).
Recall that \( \log_b 1 = 0 \) for any valid base \( b \), because \( b^0 = 1 \).
Therefore, the expression simplifies to \( \log_5 1 = 0 \).
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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_b(a) = c means b^c = a. Understanding this definition helps in interpreting and simplifying logarithmic expressions.
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7:30
Logarithms Introduction
Logarithm of the Base Itself
The logarithm of a base raised to itself, such as log_b(b), always equals 1 because b^1 = b. This property simplifies expressions like log7(7) to 1, which is crucial for evaluating nested logarithms.
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7:30Logarithms Introduction
Evaluating Nested Logarithms
Nested logarithms involve one logarithm inside another, like log5(log7(7)). To evaluate, simplify the inner logarithm first, then apply the outer logarithm. This stepwise approach avoids calculator use and clarifies the expression.
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5:14Evaluate Logarithms
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