Domanda del libro di testo
In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
logb (xy)5 = (logb x + logb y)5
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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 126
In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
Guida verificata passo dopo passo1
Recall the logarithm property for division: \( \frac{\log_b A}{\log_b B} \) is not equal to \( \log_b A - \log_b B \). Instead, the subtraction property applies to logarithms of products or quotients inside the log, such as \( \log_b \frac{A}{B} = \log_b A - \log_b B \).
Evaluate the left side expression \( \frac{\log_7 49}{\log_7 7} \). Since \( \log_7 7 = 1 \), this simplifies to \( \log_7 49 \).
Evaluate the right side expression \( \log_7 49 - \log_7 7 \). Using the subtraction property of logarithms, this equals \( \log_7 \frac{49}{7} = \log_7 7 \).
Compare the simplified left side \( \log_7 49 \) and right side \( \log_7 7 \). Since \( 49 \neq 7 \), the two sides are not equal, so the original statement is false.
To make the statement true, replace the division on the left side with subtraction inside the logarithm: \( \frac{\log_7 49}{\log_7 7} \) should be replaced by \( \log_7 \frac{49}{7} \), so the true statement is \( \log_7 \frac{49}{7} = \log_7 49 - \log_7 7 \).
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Properties of Logarithms
Logarithms have specific properties that simplify expressions, such as the product, quotient, and power rules. Understanding these properties helps in manipulating and evaluating logarithmic expressions correctly.
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Change of Base Property
Quotient Rule for Logarithms
The quotient rule states that the logarithm of a quotient is the difference of the logarithms: log_b(A/B) = log_b(A) - log_b(B). This rule is essential for rewriting and simplifying expressions involving division inside a logarithm.
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3:49Product, Quotient, and Power Rules of Logs
Evaluating Logarithms with Known Bases and Arguments
Evaluating logarithms like log7 49 and log7 7 involves recognizing powers of the base: 49 = 7^2 and 7 = 7^1. This allows simplification to numerical values, which is crucial for verifying the truth of logarithmic statements.
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5:14Evaluate Logarithms
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