Domanda del libro di testo
Graph f(x) = (1/2)x and g(x) = log1/2 x in the same rectangular coordinate system.
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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 45
Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. e4x+5e2x−24=0
Guida verificata passo dopo passo1
Start by recognizing that the equation involves exponential expressions with different exponents: \(e^{4x} + 5e^{2x} - 24 = 0\). Notice that \(e^{4x}\) can be rewritten as \((e^{2x})^2\) to simplify the equation.
Make a substitution to turn the equation into a quadratic form. Let \(u = e^{2x}\). Then the equation becomes \(u^2 + 5u - 24 = 0\).
Solve the quadratic equation \(u^2 + 5u - 24 = 0\) using the quadratic formula: \(u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=1\), \(b=5\), and \(c=-24\).
After finding the values of \(u\), substitute back \(u = e^{2x}\) and solve for \(x\) by taking the natural logarithm: \(2x = \ln(u)\), so \(x = \frac{1}{2} \ln(u)\).
Evaluate the logarithmic expressions using a calculator to find decimal approximations of \(x\), rounding to two decimal places as required.
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Exponential Equations
Exponential equations involve variables in the exponent, such as e^(4x). Solving these requires rewriting the equation to isolate the exponential term or transform it into a quadratic form, enabling the use of algebraic methods to find the variable.
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5:47
Solving Exponential Equations Using Logs
Substitution Method
When an exponential equation contains terms like e^(4x) and e^(2x), substitution can simplify it. For example, letting u = e^(2x) transforms the equation into a quadratic in u, which can be solved using factoring or the quadratic formula.
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04:03Choosing a Method to Solve Quadratics
Logarithms and Their Properties
Logarithms are the inverse of exponentials and are used to solve for variables in exponents. After isolating the exponential expression, applying natural logarithms (ln) or common logarithms (log) helps find the exact solution, which can then be approximated with a calculator.
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5:36Change of Base Property
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