Solving Exponential Equations In Quadratic Form - Using Logarithms, With e
TLDRThis video tutorial delves into solving exponential equations in quadratic form. It illustrates the process through examples, starting with the equation e^(2x) - 5e^x + 6 = 0. The method involves factoring by substitution, setting a variable 'a' equal to e^x, and then solving the resulting trinomial equation. The video demonstrates finding the values of 'a' and subsequently solving for 'x' using natural logarithms. It emphasizes that e^x can never be negative, eliminating certain solutions. Additional examples are provided, including e^(2x) - 3e^x - 28 = 0 and e^(4x) - 14e^(2x) + 48 = 0, with solutions found by applying the same technique. The video concludes by noting that taking square roots of both sides can introduce extraneous solutions, but these are typically discarded due to the nature of exponential functions. The tutorial is both informative and engaging, offering a clear understanding of solving exponential equations.
Takeaways
- ๐ The video focuses on solving exponential equations in quadratic form.
- ๐ The example equation e^(2x) - 5e^x + 6 = 0 is used to demonstrate the solving process.
- ๐ Factoring and substitution are key techniques used to solve the equation, with 'a' substituting e^x.
- ๐ข The numbers that multiply to the constant term and add up to the middle coefficient are identified to factor the trinomial.
- ๐ Setting each factor equal to zero allows solving for 'a', which is then substituted back to find e^x.
- ๐ The natural logarithm is used to isolate 'x' by taking ln on both sides of the equation.
- ๐ซ It's emphasized that e^x cannot be negative, so equations where e^x equals a negative number have no solution.
- ๐ The process is repeated with different equations to demonstrate the consistency of the method.
- ๐ The video also covers the case where e^(2x) is set equal to different positive numbers and how to solve for 'x'.
- ๐ข The solutions involve taking the natural logarithm and manipulating the exponent to solve for 'x'.
- ๐ The video concludes with a reminder that all methods lead to the same solutions, even when using square roots.
Q & A
What is the main topic of the video?
-The main topic of the video is solving exponential equations in quadratic form.
What is the first example equation given in the video?
-The first example equation is e^(2x) - 5e^x + 6 = 0.
What substitution is used to simplify the first example equation?
-The substitution used is to let 'a' be equal to e^x, which simplifies e^(2x) to a^2.
What is the trinomial obtained after the substitution in the first example?
-The trinomial obtained is a^2 - 5a + 6 = 0.
How do you find the factors of the trinomial a^2 - 5a + 6?
-You find the factors by looking for two numbers that multiply to 6 and add up to -5, which are -2 and -3.
What are the solutions for 'a' in the first example?
-The solutions for 'a' are a = 2 and a = 3.
How do you find the value of x when e^x = 2?
-You find the value of x by taking the natural log of both sides, resulting in x = ln(2).
What is the second example equation provided in the video?
-The second example equation is e^(2x) - 3e^x - 28 = 0.
Why is e^x never equal to a negative number?
-e^x is never equal to a negative number because the exponential function e^x is always positive and never crosses the x-axis.
What is the process to solve e^(2x) = 8?
-The process involves taking the natural log of both sides and then dividing by 2, resulting in x = (1/2)ln(8).
What are the solutions for x in the equation e^(4x) - 14e^(2x) + 48 = 0?
-The solutions for x are x = (1/2)ln(8) and x = (1/2)ln(6).
Why can't we use the square root method for e^(2x) = 8 to find additional solutions?
-We can't use the square root method because it would lead to e^x = -โ8, which is not possible since e^x can never be negative.
Outlines
๐งฎ Solving Exponential Equations Using Substitution
This paragraph explains how to solve exponential equations in quadratic form using substitution. The example equation \( e^{2x} - 5e^x + 6 = 0 \) is given, and the substitution \( a = e^x \) is used to transform the equation into a quadratic \( a^2 - 5a + 6 = 0 \). Factoring the quadratic leads to solutions for \( a \), which are then converted back to solve for \( x \) by taking the natural logarithm. The solutions \( x = \ln 2 \) and \( x = \ln 3 \) are derived.
๐ Example of Exponential Equation with No Solution
This paragraph provides another example of solving exponential equations: \( e^{2x} - 3e^x - 28 = 0 \). Using the substitution \( a = e^x \), the equation is transformed into \( a^2 - 3a - 28 = 0 \). Factoring leads to solutions \( a = -4 \) and \( a = 7 \). The equation \( e^x = 7 \) is solved to find \( x = \ln 7 \), but \( e^x = -4 \) has no solution because the exponential function cannot equal a negative number. This illustrates that exponential functions can never be negative.
๐ Final Example and Additional Solution Method
In this paragraph, a more complex exponential equation \( e^{4x} - 14e^{2x} + 48 = 0 \) is solved using substitution. Setting \( a = e^{2x} \) transforms the equation to \( a^2 - 14a + 48 = 0 \), which is then factored to find \( a = 8 \) and \( a = 6 \). Solving \( e^{2x} = 8 \) and \( e^{2x} = 6 \) yields solutions \( x = \frac{\ln 8}{2} \) and \( x = \frac{\ln 6}{2} \). The paragraph also explores using the square root to find \( e^x = \pm \sqrt{8} \), concluding that only the positive root is valid, reinforcing that exponential functions cannot be negative.
Mindmap
Keywords
๐กExponential Equations
๐กQuadratic Form
๐กFactoring
๐กSubstitution
๐กNatural Logarithm
๐กLeading Coefficient
๐กMiddle Coefficient
๐กConstant Term
๐กHorizontal Asymptote
๐กNatural Logarithm Properties
๐กSquare Root
Highlights
Introduction to solving exponential equations in quadratic form.
Example equation: e^(2x) - 5e^x + 6 = 0.
Using substitution with a = e^x to simplify the equation.
Factoring the resulting trinomial equation a^2 - 5a + 6 = 0.
Finding factors that multiply to 6 and add to -5: -2 and -3.
Setting each factor equal to zero to find values of a.
Solving for x when a = e^x: x = ln(2) and x = ln(3).
Demonstration of using natural log to solve for x.
New example equation: e^(2x) - 3e^x - 28 = 0.
Factoring the equation a^2 - 3a - 28 = 0.
Identifying factors of -28 that add up to -3: -4 and 7.
Realization that e^x cannot equal a negative number.
Explanation of the properties of exponential functions.
Third example equation: e^(4x) - 14e^(2x) + 48 = 0.
Using substitution with a = e^(2x) to form a^2 - 14a + 48 = 0.
Finding factors of 48 that add up to -14: -8 and -6.
Solving for x when e^(2x) = 8 and e^(2x) = 6.
Using natural log to solve for x in the equation e^(2x) = 8.
Solving for x in the equation e^(2x) = 6 and finding x = (1/2)ln(6).
Exploring alternative solutions involving square roots.
Clarification that e^x cannot be negative, eliminating ln(-4).
Final solutions for the equation: x = (1/2)ln(8) and x = (1/2)ln(6).
Transcripts
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