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. e^2x−3e^x+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 \(u = e^{x}\).
2. Rewrite the equation in terms of \(u\): since \(e^{2x} = (e^{x})^2\), the equation becomes \(u^2 - 3u + 2 = 0\).
3. Solve the quadratic equation \(u^2 - 3u + 2 = 0\) by factoring or using the quadratic formula to find the values of \(u\).
4. After finding the solutions for \(u\), substitute back \(u = e^{x}\) and solve for \(x\) by taking the natural logarithm: \(x = \ln(u)\).
5. 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.

Key Concepts

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.

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.

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.