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Magazine Chapter 5: Problem 29
Solve the exponential equation algebraically. Then check using a graphing calculator. Round to three decimal places, if appropriate. $$2 e^{x}=5-e^{-x}$$
Short Answer
Expert verified
x \approx 0.576
Step by step solution
01
- Rewrite the equation
Rewrite the given equation so that all the terms are on one side of the equation: \[ 2e^x + e^{-x} - 5 = 0 \]
02
- Substitute to simplify
Let \( y = e^x \). Therefore, \( e^{-x} = \frac{1}{y} \). Replace \( e^x \) and \( e^{-x} \) in the equation: \[ 2y + \frac{1}{y} - 5 = 0 \]
03
- Multiply to clear the fraction
Multiply every term in the equation by \( y \) to clear the fraction: \[ 2y^2 + 1 - 5y = 0 \]
04
- Form a quadratic equation
Rearrange the equation to match the standard form of a quadratic equation \( ay^2 + by + c = 0 \): \[ 2y^2 - 5y + 1 = 0 \]
05
- Solve the quadratic equation
Use the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 2 \), \( b = -5 \), and \( c = 1 \): \[ y = \frac{5 \pm \sqrt{(-5)^2 - 4(2)(1)}}{4} \] \[ y = \frac{5 \pm \sqrt{25 - 8}}{4} \] \[ y = \frac{5 \pm \sqrt{17}}{4} \]
06
- Find the solution for x
Since \( y = e^x \), find \( x \) by taking the natural logarithm of both solutions: \[ e^x = \frac{5 + \sqrt{17}}{4} \] \[ x = \ln \left( \frac{5 + \sqrt{17}}{4} \right) \] or \[ e^x = \frac{5 - \sqrt{17}}{4} \] Since \( \frac{5 - \sqrt{17}}{4} \approx -0.281 \) is not feasible for a positive exponential function, we have: \[ x = \ln \left( \frac{5 + \sqrt{17}}{4} \right) \]
07
- Calculate the decimal value
Use a calculator to evaluate the natural logarithm to three decimal places: \[ x \approx \ln \left( 1.780 \right) \approx 0.576 \]
08
- Check with a graphing calculator
Plot the functions \( y = 2e^x \) and \( y = 5 - e^{-x} \) on a graphing calculator. Verify the intersection point is approximately at \( x \approx 0.576 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Exponential Functions
Exponential functions are mathematical expressions in which a variable appears in the exponent. These functions can model growth or decay processes, such as population growth or radioactive decay. The general form of an exponential function is \(f(x) = a b^x\), where
- \(a\) is the initial value,
- \(b\) is the base, > 0 and \(b eq 1\),
- \(x\) is the exponent or power.
Quadratic Equations
A quadratic equation is a polynomial equation of the second degree. The standard form is \(ax^2 + bx + c = 0\), where
- \(a\),
- \(b\), and
- \(c\) are constants,
- If \(b^2 - 4ac > 0\), there are two distinct real roots,
- If \(b^2 - 4ac = 0\), there is one real root (a repeated root), and
- If \(b^2 - 4ac < 0\), the roots are complex (not real).
Natural Logarithm
The natural logarithm is the inverse of the exponential function with base \(e\). Denoted as \(\ln(x)\), it answers the question: 'To what power must \(e\) be raised to obtain \(x\)? Some important properties of the natural logarithm include:
- \(\ln(1) = 0\), since \(e^0 = 1\),
- \(\ln(e) = 1\), since \(e^1 = e\), and
- \(\ln(ab) = \ln(a) + \ln(b)\), which shows the logarithm of a product is the sum of the logarithms.
