Skip to main content
Ch. 4 - Inverse, Exponential, and Logarithmic Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 4 - Inverse, Exponential, and Logarithmic FunctionsProblem 37
Chapter 5, Problem 37

Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. 2e2x + ex = 6

Verified step by step guidance
1
Recognize that the equation involves exponential expressions with the same base, \(e\). The equation is \(2e^{2x} + e^x = 6\).
Rewrite \(e^{2x}\) as \((e^x)^2\) to transform the equation into a quadratic form in terms of \(e^x\). This gives \(2(e^x)^2 + e^x = 6\).
Let \(u = e^x\). Substitute into the equation to get a quadratic equation: \(2u^2 + u = 6\).
Rearrange the quadratic equation to standard form: \(2u^2 + u - 6 = 0\).
Solve the quadratic equation for \(u\) using the quadratic formula \(u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=2\), \(b=1\), and \(c=-6\). After finding the values of \(u\), substitute back \(u = e^x\) and solve for \(x\) by taking the natural logarithm: \(x = \ln(u)\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

Key Concepts

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

Exponential Functions

Exponential functions involve variables in the exponent, such as e^x, where e is Euler's number (~2.718). Understanding their properties, like growth behavior and how to manipulate expressions with exponents, is essential for solving equations involving terms like e^x and e^{2x}.
Recommended video:
6:13
Exponential Functions

Substitution Method for Solving Equations

When an equation contains terms like e^{2x} and e^x, substitution can simplify it by letting a new variable represent e^x. This transforms the equation into a quadratic form, making it easier to solve using algebraic methods before back-substituting to find x.
Recommended video:
Guided course
5:48
Solving Systems of Equations - Substitution

Solving Quadratic Equations

After substitution, the equation often becomes quadratic, requiring methods like factoring, completing the square, or the quadratic formula to find solutions. Understanding how to solve quadratics is crucial to determine the values of the substituted variable and ultimately solve for x.
Recommended video:
06:08
Solving Quadratic Equations by Factoring
Related Practice
Textbook Question

Graph each function. See Example 2. ƒ(x) = 2|x|

Textbook Question

Suppose that water from a wetland area is sampled and found to have the given hydronium ion concentration. Determine whether the wetland is a rich fen, a poor fen, or a bog. 2.49×\(\times\)10-5

Textbook Question

Solve each equation. log1/2 (x+3) = -4

Textbook Question

Solve each equation. log9 x = 5/2

Textbook Question

Solve each equation. log1/3 (x+6) = -2

2
views
Textbook Question

Find the [H3O+] for each substance with the given pH. Write answers in scientific notation to the nearest tenth. beer, 4.8