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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblema 7
Capitolo 5, Problema 7

Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 42x−1=64

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1
Recognize that the equation is \(4^{2x - 1} = 64\). The goal is to express both sides as powers of the same base.
Rewrite the base 4 and 64 as powers of 2, since \(4 = 2^2\) and \(64 = 2^6\).
Substitute these expressions back into the equation to get \((2^2)^{2x - 1} = 2^6\).
Use the power of a power property: \((a^m)^n = a^{m \cdot n}\), so rewrite the left side as \(2^{2(2x - 1)}\).
Since the bases are the same (base 2), set the exponents equal: \(2(2x - 1) = 6\). Then solve this linear equation for \(x\).

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Exponential Equations

An exponential equation is one in which variables appear as exponents. Solving these equations often involves rewriting both sides with the same base to compare the exponents directly.
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Expressing Numbers as Powers of the Same Base

To solve exponential equations, rewrite each side as a power of the same base. For example, 64 can be expressed as 4³ since 4³ = 64. This allows the exponents to be set equal to each other.
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Equating Exponents

Once both sides of an equation have the same base, their exponents can be set equal. This transforms the problem into a simpler algebraic equation, which can be solved using standard methods.
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