E 2x 5e X 6 0

Decoding the Enigma: Exploring the Equation "e^(2x) = 5e^(x) - 6"

Let's face it, staring at an equation like "e^(2x) = 5e^(x) - 6" can feel like staring into the abyss. But what if I told you this seemingly complex equation holds the key to understanding a wide range of phenomena, from population growth to radioactive decay? This isn't just abstract math; it's a powerful tool with real-world applications. Let's dive in and unravel the mystery behind this exponential equation.

1. Unveiling the Exponential Nature:

At its heart, this equation deals with exponential functions, those remarkable curves that describe phenomena exhibiting rapid growth or decay. The base "e," also known as Euler's number (approximately 2.718), is a fundamental constant in mathematics, appearing everywhere from compound interest calculations to the modeling of natural processes. The equation itself represents a balance between two exponential terms: e^(2x) and 5e^(x). Understanding this balance is key to solving it. Imagine, for example, two competing bacterial colonies: one growing exponentially at a rate proportional to e^(2x) and another growing at a rate proportional to 5e^(x). Our equation describes the point where their growth rates are equal.

2. Transforming the Equation: A Clever Substitution:

Directly solving for 'x' in the equation e^(2x) = 5e^(x) - 6 might seem daunting. But here’s where a clever substitution comes in handy. Notice that e^(2x) is simply (e^(x))^2. Let's substitute 'y' for e^(x). Our equation transforms into a much more manageable quadratic equation: y² = 5y - 6. This is a significant simplification, taking us from the realm of exponential functions to the familiar territory of quadratic equations, solvable using standard techniques.

3. Solving the Quadratic and Back to Exponential Form:

Rearranging our quadratic equation, we get y² - 5y + 6 = 0. This factors nicely into (y - 2)(y - 3) = 0, giving us two possible solutions for 'y': y = 2 and y = 3. Remember, y = e^(x). Therefore, we have two separate exponential equations to solve: e^(x) = 2 and e^(x) = 3.

4. Extracting the Solutions for 'x': The Power of Natural Logarithms:

To solve for 'x', we employ the natural logarithm (ln), the inverse function of the exponential function with base 'e'. Taking the natural logarithm of both sides of e^(x) = 2 and e^(x) = 3, we get:

x = ln(2) and x = ln(3)

These are our two solutions. In decimal form, x ≈ 0.693 and x ≈ 1.099. These solutions represent the specific points where the two exponential terms in the original equation achieve balance. In our bacterial colony example, these would be the times at which the growth rates of the two colonies are equal.

5. Real-World Applications and Beyond:

The techniques used to solve this equation are applicable across a range of real-world problems. Examples include:

Radioactive decay: Modeling the decay of radioactive isotopes.
Population dynamics: Analyzing population growth and decline in ecology.
Financial modeling: Calculating compound interest and investment growth.
Chemical kinetics: Describing reaction rates in chemical processes.

Understanding and solving equations like this empowers us to model and predict behavior in these diverse fields.

Conclusion:

Solving "e^(2x) = 5e^(x) - 6" might have seemed intimidating at first, but by employing strategic substitution and leveraging the properties of exponential and logarithmic functions, we've successfully found its solutions. This exercise highlights the power of mathematical tools in tackling complex real-world problems, demonstrating the interconnectedness between seemingly disparate areas of study. The ability to analyze and solve exponential equations is a cornerstone of many scientific and engineering disciplines.

Expert-Level FAQs:

1. What if the quadratic equation resulting from the substitution doesn't factor easily? Use the quadratic formula: x = [-b ± √(b² - 4ac)] / 2a.

2. Can this equation have complex solutions? No, in this particular case, the discriminant (b² - 4ac) of the quadratic is positive, guaranteeing real solutions. However, other exponential equations can lead to complex solutions.

3. How would the solution approach change if the equation was e^(3x) = 7e^(2x) - 12? A similar substitution would work, leading to a cubic equation instead of a quadratic. Solving cubic equations can be more involved, potentially requiring numerical methods.

4. Are there limitations to the use of this substitution method? Yes, the method relies on the ability to express higher powers of exponentials as powers of lower-order exponentials. This isn't always feasible.

5. How can I verify my solutions? Substitute your calculated values of 'x' back into the original equation to confirm that both sides are equal. Using a calculator or software to evaluate the exponential terms is recommended for accuracy.

Search Results:

Find the solutions of the exponential equation e^(2x) - 5e^(x) + 6 = 0 ... Find the solutions of the exponential equation e 2 x − 5 e x + 6 = 0. Equation in which the variable which we have to solve is given in the exponent is called exponential equation. To...

e^2x-5e^x+6=0 - Symbolab Detailed step by step solution for e^2x-5e^x+6=0. Line Graph Calculator Exponential Graph Calculator Quadratic Graph Calculator Sin graph Calculator More...

Solve for x: e^{2x}-5e^x+6=0 | Homework.Study.com This problem involves the substitution of variables to solve the given equation. Since solving for variables which are present as exponential powers, we need to substitute them and simplify the...

e^{2x}-5e^x+6=0 - Symbolab x^{2}-x-6=0 -x+3\gt 2x+1 ; line\:(1,\:2),\:(3,\:1) f(x)=x^3 ; prove\:\tan^2(x)-\sin^2(x)=\tan^2(x)\sin^2(x) \frac{d}{dx}(\frac{3x+9}{2-x}) (\sin^2(\theta))' \sin(120)

Solved: e^(2x)-5e^x+6=0 [algebra] - gauthmath.com Answer: x = ln ⁡ 2 x=\ln 2 x = ln 2 or x = ln ⁡ 3 x=\ln 3 x = ln 3. Explanation: Solve the equations:

Question: Solve the equation e^2x-5e^x+6=0 - Chegg Our expert help has broken down your problem into an easy-to-learn solution you can count on. Here’s the best way to solve it. Not the question you’re looking for? Post any question and get …

Solve e^2x-5e^x=0 | Microsoft Math Solver How do I solve for \displaystyle{x} ? \displaystyle{2}{e}^{{{2}{x}}}-{5}{e}^{{x}}-{3}={0} This problem is about the natural logarithm function.

50 | algebra MCQ - Practice Quiz with Solutions | Question 31 Solve for: e2x - 5ex + 6 = 0 A Computer Science portal for geeks. It contains well written, well thought and well explained computer science and programming articles, quizzes and practice/competitive programming/company interview Questions.

Solve the Exponential Equation e^(2x)-5e^x+6=0 - Step-by-Step … Use substitution, factoring, and logarithms to solve the exponential equation e^(2x)-5e^x+6=0.

solvefor x,e^{2x}-5e^x+6=0 - Symbolab x = 1.09861…,x = 0.69314… What is the answer to solvefor x,e^ {2x}-5e^x+6=0 ?

Question: e^2x-5e^x+6=0 - Chegg There’s just one step to solve this. Graph each side of... Not the question you’re looking for? Post any question and get expert help quickly. 106,400+ students just like you got their calculus questions answered this last week.

Question: Find the solutions of the exponential equation e^(2x)-5e^(x)+6=0 Find the solutions of the exponential equation e^(2x)-5e^(x)+6=0 Your solution’s ready to go! Our expert help has broken down your problem into an easy-to-learn solution you can count on.

Solve e^2x-5e^x+6=0 | Microsoft Math Solver Use e2x = (ex)2 to see that this equation is quadratic in ex . Explanation: e2x −6ex+8 = 0 ... \left. \begin {cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end {cases} \right. Solve your math problems using our free math solver with step-by-step solutions. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more.

e^{2x}-5e^x-6=0 - Symbolab Detailed step by step solution for e^{2x}-5e^x-6=0. Solutions. Integral Calculator Derivative Calculator Algebra Calculator Matrix Calculator More... Graphing. Line Graph Exponential Graph Quadratic Graph Sin graph More... Calculators. ... e 2 x − 5 e x − 6 = 0

Solve for x e^(2x)-5e^x+6=0 | Mathway Set ex −3 e x - 3 equal to 0 0 and solve for x x. Tap for more steps... Set ex −2 e x - 2 equal to 0 0 and solve for x x. Tap for more steps... The final solution is all the values that make (ex −3)(ex −2) = 0 (e x - 3) (e x - 2) = 0 true. The result can be shown in …

Solve: ${e^{2x}} - 5{e^x} + 6 = 0$. - Vedantu So, the given equation ${e^{2x}} - 5{e^x} + 6 = 0$becomes $t{}^2 - 5t + 6 = 0$ . Now, considering equation $t{}^2 - 5t + 6 = 0$, the equation can be solved by various methods such as completing the square method, splitting the middle term and using the quadratic formula.

e^{2x}+5e^x-6=0 - Symbolab What is the answer to e^ {2x}+5e^x-6=0 ?

How do you solve e^(2x)-5e^x+6=0? - Socratic 13 Jan 2017 · How do you solve e^ (2x)-5e^x+6=0? We have: e^ (2 x) - 5 e^ (x) + 6 = 0. Using the laws of exponents: Rightarrow (e^ (x))^ (2) - 5 e^ (x) + 6 = 0. Let's factorise the equation: Rightarrow (e^ (x))^ (2) - 2 e^ (x) - 3 e^ (x) + 6 = 0. Rightarrow e^ (x) (e^ (x) - 2) - 3 (e^ (x) - 2) = 0. Rightarrow (e^ (x) - 2) (e^ (x) - 3) = 0.

e^{2x}+5e^x-6=0 - Symbolab x^{2}-x-6=0 -x+3\gt 2x+1 ; line\:(1,\:2),\:(3,\:1) f(x)=x^3 ; prove\:\tan^2(x)-\sin^2(x)=\tan^2(x)\sin^2(x) \frac{d}{dx}(\frac{3x+9}{2-x}) (\sin^2(\theta))' \sin(120)

How do you solve e^(2x)-5e^x+6=0? - Socratic 21 Apr 2016 · Let # u=e^x# Then we have. #u^2-5u+6=0# #(u-3)(u-2)=0# #u-3=0 or u-2=0# #e^x-3=0 or e^x-2=0# #e^x=3 or e^x=2# #lne^x=ln3 or lne^x = ln 2# #x ln e=ln3 or xlne=ln2# #x=ln3 or x=ln2#

E 2x 5e X 6 0

Decoding the Enigma: Exploring the Equation "e^(2x) = 5e^(x) - 6"

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