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. 3e5x=1977
Table of contents
- 0. Review of Algebra4h 18m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations1h 43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 5m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 22m
- 10. Combinatorics & Probability1h 45m
6. Exponential & Logarithmic Functions
Solving Exponential and Logarithmic Equations
Problem 35
Textbook Question
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. e2x - 6ex + 8 = 0
Verified step by step guidance1
Recognize that the equation involves exponential expressions with the same base \(e\). To simplify, let \(y = e^{x}\). Then, \(e^{2x} = (e^{x})^2 = y^2\).
Rewrite the original equation \(e^{2x} - 6e^{x} + 8 = 0\) in terms of \(y\) as \(y^2 - 6y + 8 = 0\).
Solve the quadratic equation \(y^2 - 6y + 8 = 0\) using factoring or the quadratic formula. This will give you the possible values for \(y\).
Recall that \(y = e^{x}\), so for each solution \(y_i\), solve the equation \(e^{x} = y_i\) by taking the natural logarithm of both sides: \(x = \ln(y_i)\).
Evaluate the logarithms to find the values of \(x\). Since the problem asks for decimal answers correct to the nearest thousandth, approximate the logarithms accordingly.
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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^x. Solving these requires understanding how to manipulate and isolate the exponential expression, often by substitution or applying logarithms.
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Substitution Method for Quadratic Form
When an equation contains terms like e^{2x} and e^x, it can be treated as a quadratic by substituting u = e^x. This transforms the equation into a quadratic form, which can be solved using factoring or the quadratic formula.
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Choosing a Method to Solve Quadratics
Exact vs. Approximate Solutions
Exact solutions are expressed in terms of constants and variables without rounding, while approximate solutions are numerical values rounded to a specified decimal place. Understanding when to use each is important for interpreting answers correctly.
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