Textbook Question
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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Table of contents
- 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
Chapter 5, Problem 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.
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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
Related Practice
Textbook Question
In Exercises 109–112, find the domain of each logarithmic function. f(x) = log[(x+1)/(x-5)]
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Textbook Question
In Exercises 109–112, find the domain of each logarithmic function. f(x) = ln (x² - x − 2)
Textbook Question
In Exercises 105–108, evaluate each expression without using a calculator. log (ln e)
Textbook Question
If log 3 = A and log 7 = B, find log7 (9) in terms of A and B.
Textbook Question
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.