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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblem 43
Chapter 5, Problem 43

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. e2x−3ex+2=0

Verified step by step guidance
1
Recognize that the equation \(e^{2x} - 3e^{x} + 2 = 0\) is an exponential equation that can be transformed into a quadratic form by substituting \(u = e^{x}\).
Rewrite the equation in terms of \(u\): since \(e^{2x} = (e^{x})^2\), the equation becomes \(u^2 - 3u + 2 = 0\).
Solve the quadratic equation \(u^2 - 3u + 2 = 0\) by factoring or using the quadratic formula to find the values of \(u\).
After finding the solutions for \(u\), substitute back \(u = e^{x}\) and solve for \(x\) by taking the natural logarithm: \(x = \ln(u)\).
Express the solution set in terms of natural logarithms and then use a calculator to approximate the decimal values of \(x\) to two decimal places.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Exponential Equations

Exponential equations involve variables in the exponent, such as e^(2x). Solving these requires rewriting the equation to isolate the exponential term or substituting to form a quadratic equation, enabling the use of algebraic methods to find the variable.
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Solving Exponential Equations Using Logs

Logarithms and Their Properties

Logarithms are the inverse operations of exponentials, allowing us to solve for variables in exponents. Natural logarithms (ln) use base e, while common logarithms use base 10. Applying logarithms helps isolate the variable after rewriting the equation.
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Change of Base Property

Quadratic Equation Techniques

When an exponential equation can be rewritten as a quadratic form, techniques like factoring or the quadratic formula are used to find solutions. Recognizing this form simplifies solving complex exponential equations by treating e^x as a single variable.
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Introduction to Quadratic Equations
Related Practice
Textbook Question

Graph f(x) = 4x and g(x) = log4 x in the same rectangular coordinate system.

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Textbook Question

Evaluate each expression without using a calculator. 8log8198^{\(\log\)_8 19}

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Textbook Question

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. ln x + ln 7

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Textbook Question

The figure shows the graph of f(x) = ex. In Exercises 35-46, use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine graphs. each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn h(x) = e2x + 1

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Textbook Question

The figure shows the graph of f(x) = ex. In Exercises 35-46, use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine graphs. each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn g(x) = 2ex

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Textbook Question

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. 5(2x+3)=3(x−1)

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