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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 17
Capitolo 5, Problema 17

Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 4x=1/√2

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Rewrite the equation \(4^{x} = \frac{1}{\sqrt{2}}\) by expressing both sides as powers of the same base. Note that 4 can be written as \$2^{2}$ and \(\sqrt{2}\) can be written as \(2^{\frac{1}{2}}\).
Rewrite the left side as \(\left(2^{2}\right)^{x}\) and the right side as \(\frac{1}{2^{\frac{1}{2}}}\).
Use the property of exponents \(\left(a^{m}\right)^{n} = a^{mn}\) to simplify the left side to \$2^{2x}$.
Rewrite the right side \(\frac{1}{2^{\frac{1}{2}}}\) as \(2^{-\frac{1}{2}}\) using the negative exponent rule \(\frac{1}{a^{m}} = a^{-m}\).
Since the bases are the same (base 2), set the exponents equal: \(2x = -\frac{1}{2}\). Then solve for \(x\) by dividing both sides by 2.

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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 so that the exponents can be set equal, simplifying the problem to solving a linear equation.
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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 powers of the same base. For example, 1/√2 can be expressed as 2 raised to a negative fractional exponent, allowing the equation to be compared directly by their exponents.
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Equating Exponents

Once both sides of an exponential equation have the same base, the exponents can be set equal to each other. This step transforms the problem into a simpler algebraic equation, which can be solved using standard algebraic methods.
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Rational Exponents
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