How to solve exponential equations with unequal coefficients?

  • Thread starter Ali Asadullah
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In summary, when solving equations of the form e^ax+e^bx=k, where a=2b, we can divide through by e^{bx} to simplify the equation. However, if a is not equal to 2b, the equation becomes more difficult and there is no general method for solving it. The website www.wolframalpha.com can provide helpful solutions and step-by-step explanations for these types of problems.
  • #1
Ali Asadullah
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We know how to solve problems like e^ax+e^bx=k,
when a=2b. But how to solve equations of this type when a is not equal to 2b?
 
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  • #2
Divide through by e^{bx} so that the equation becomes e^{(a-b)x}- ke^{-bx}+ 1= (e^{bx})^{(a-b)/b}- ke^{-bx}+ 1 0. Now let y= e^{-bx} and the equation is
y^{(a-b)/b}- ky+ 1= 0. That will be a polynomial equation if and only if (a-b)/b is a positive integer. Other wise it is a more difficult equation. There is no general method of solving such equations, not even all polynomial equations.
 
  • #3
go to www.wolframalpha.com

it is very helpful with these kind of problems !

it sometimes also shows u step by step, how they got the answer, depending on question
 

1. What is an exponential equation?

An exponential equation is an equation in which the variable appears in the exponent. It is typically written in the form y = ab^x, where a and b are constants and x is the variable.

2. How do you solve an exponential equation?

To solve an exponential equation, you can use logarithms. Take the logarithm of both sides of the equation and then use algebraic techniques to isolate the variable. Remember to check for extraneous solutions, as some logarithms may produce complex solutions.

3. What are the properties of exponential equations?

Exponential equations have several properties. One important property is the power rule, which states that for any real numbers a and b, and any nonzero real number c, (ab)^c = a^c * b^c. Another property is the product rule, which states that a^x * a^y = a^(x+y). Additionally, exponential equations have a constant base, meaning that the variable is raised to a fixed power.

4. What are some real-world applications of exponential equations?

Exponential equations have many real-world applications, such as calculating population growth, compound interest, and radioactive decay. They can also be used to model data that increases or decreases at a constant rate, such as the spread of a virus or the growth of a bacteria colony.

5. What are some common mistakes when solving exponential equations?

One common mistake when solving exponential equations is forgetting to use logarithms to isolate the variable. It is also important to check for extraneous solutions, as mentioned before. Another mistake is not paying attention to the properties of exponential equations, such as the power and product rules. Lastly, it is important to check your final solution to make sure it makes sense in the context of the problem.

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