Domanda del libro di testo
In Exercises 13–15, write each equation in its equivalent exponential form. 1/2 = log49 7
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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 13
Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 31-x=1/27
Guida verificata passo dopo passo1
Recognize that the equation is \(3^{1-x} = \frac{1}{27}\). The goal is to express both sides with the same base.
Recall that \(27\) can be written as a power of \(3\) because \(27 = 3^3\). Therefore, rewrite the right side as \(\frac{1}{3^3}\).
Use the property of exponents that \(\frac{1}{a^n} = a^{-n}\) to rewrite \(\frac{1}{3^3}\) as \(3^{-3}\).
Now the equation is \(3^{1-x} = 3^{-3}\). Since the bases are the same and the function is one-to-one, set the exponents equal: \(1 - x = -3\).
Solve the equation \(1 - x = -3\) for \(x\) by isolating \(x\) on one side.
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Exponential Equations
An exponential equation is one in which variables appear as exponents. Solving these equations often involves rewriting expressions so that both sides have the same base, allowing the exponents to be set equal to each other.
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Solving Exponential Equations Using Logs
Expressing Numbers as Powers of the Same Base
To solve exponential equations, it is essential to rewrite each side as a power of the same base. For example, 1/27 can be expressed as 3 to the power of -3, since 27 = 3^3 and the reciprocal is 3^-3.
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Equating Exponents
Once both sides of an equation have the same base, the exponents can be set equal to each other because if a^m = a^n and a ≠ 0 or 1, then m = n. This step simplifies the problem to solving a linear equation in the exponent.
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04:06Rational Exponents
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