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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 19
Capitolo 5, Problema 19

Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 8(x+3)=16(x−1)

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Identify the bases on both sides of the equation: \(8^{(x+3)} = 16^{(x-1)}\). Notice that both 8 and 16 can be expressed as powers of 2.
Rewrite each base as a power of 2: \(8 = 2^3\) and \(16 = 2^4\). Substitute these into the equation to get \((2^3)^{(x+3)} = (2^4)^{(x-1)}\).
Apply the power of a power property: \((a^m)^n = a^{m \cdot n}\). This gives \(2^{3(x+3)} = 2^{4(x-1)}\).
Since the bases are the same (both are base 2), set the exponents equal to each other: \(3(x+3) = 4(x-1)\).
Solve the resulting linear equation for \(x\) by expanding both sides and isolating \(x\).

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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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Expressing Numbers as Powers of the Same Base

To solve exponential equations, it is helpful to rewrite each number as a power of a common base. For example, 8 can be written as 2³ and 16 as 2⁴, enabling the comparison of exponents when bases match.
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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, then m = n. This step transforms the problem into a simpler algebraic equation to solve for the variable.
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Rational Exponents
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The graph of an exponential function is given. Select the function for each graph from the following options:

f(x)=3x,g(x)=3x−1,h(x)=3x−1,f(x)=−3x,G(x)=3−x,H(x)=−3−x.f(x) = 3^x, \(\quad\) g(x) = 3^{x-1}, \(\quad\) h(x) = 3^x - 1, \\ f(x) = -3^x, \(\quad\) G(x) = 3^{-x}, \(\quad\) H(x) = -3^{-x}.

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In Exercises 19–24, the graph of an exponential function is given. Select the function for each graph from the following options:

f(x)=3x,g(x)=3x−1,h(x)=3x−1,f(x)=−3x,G(x)=3−x,H(x)=−3−x.f(x) = 3^x, \(\quad\) g(x) = 3^{x-1}, \(\quad\) h(x) = 3^x - 1, \\ f(x) = -3^x, \(\quad\) G(x) = 3^{-x}, \(\quad\) H(x) = -3^{-x}.

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Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. lnx5\(\ln\)\(\sqrt\)[5]{x}

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Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. e(x+1)=1/e

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Write each equation in its equivalent logarithmic form. 7y = 200

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In Exercises 19–29, evaluate each expression without using a calculator. If evaluation is not possible, state the reason. log5 (1/5)

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