Domanda del libro di testo
Solve each equation.
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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 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)]
Guida verificata passo dopo passo1
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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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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5:26
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
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