Textbook Question
Solve each equation.
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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 101
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)]
Verified step by step guidance1
Recall the change of base formula for logarithms: \(\log_a(b) = \frac{\log_c(b)}{\log_c(a)}\) for any positive base \(c \neq 1\).
Apply the change of base formula to \(\log_7(3)\) using base 3: \(\log_7(3) = \frac{\log_3(3)}{\log_3(7)}\).
Since \(\log_3(3) = 1\), simplify the expression to \(\log_7(3) = \frac{1}{\log_3(7)}\).
Notice that this means \(\log_3(7) = \frac{1}{\log_7(3)}\), which matches the original equation given.
Therefore, the equation \(\log_3(7) = \frac{1}{\log_7(3)}\) is true, based on the properties of logarithms.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Logarithmic Functions and Notation
A logarithm log_b(a) answers the question: to what power must the base b be raised to get a? Understanding this notation is essential for interpreting and manipulating logarithmic equations.
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Graphs of Logarithmic Functions
Change of Base Formula
The change of base formula states that log_b(a) = 1 / log_a(b). This property allows rewriting logarithms with different bases and is key to verifying or transforming logarithmic equations.
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5:36Change of Base Property
Verifying Logarithmic Equations
To determine if a logarithmic equation is true, substitute values or apply logarithmic properties like the change of base formula. Showing work involves rewriting expressions and simplifying to confirm equality.
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5:38Verifying if Equations are Functions
Related Practice
Textbook Question
In Exercises 105–108, evaluate each expression without using a calculator. log5 (log7 7)
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Textbook Question
Evaluate or simplify each expression without using a calculator. 10log ∛x
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Textbook Question
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. log6 [4(x + 1)] = log6 (4) + log6 (x + 1)
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Textbook Question
In Exercises 101–104, write each equation in its equivalent exponential form. Then solve for x. log4x=-3
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Textbook Question
Write each equation in its equivalent exponential form. Then solve for x. log3 (x-1) = 2
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