Solving this exponential equation

  • Thread starter Emethyst
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Then, use the quadratic formula to solve for e^x. Once you have the values for e^x, you can take the natural logarithm of both sides to solve for x. In summary, to solve the equation e^x+10e^-x=7, use logarithm rules and the natural logarithm to rewrite the equation, then use the quadratic formula to solve for e^x and take the natural logarithm to find the values of x.
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Emethyst
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Homework Statement


Solve the equation e^x+10e^-x=7


Homework Equations


Logarithm rules and the natural logarithm



The Attempt at a Solution


Not really a calculus question but one I'm lost on nevertheless. I don't know how to go about solving this question at all. My first attempt was to factor out the e^x, but this got me nowhere. I know that this question has two answers (ln2 and ln5), but I cannot seem to figure out how to solve for these answers. Any help would be greatly appreciated, thanks in advance.
 
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  • #2
Multiply both sides by e^x, which will give you an equation that is quadratic in form.
 

FAQ: Solving this exponential equation

What is an exponential equation?

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

How do I 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 the properties of logarithms to solve for the variable. You may also need to use algebraic techniques to simplify the equation before taking the logarithm.

What is the difference between solving an exponential equation and a logarithmic equation?

The main difference between solving an exponential equation and a logarithmic equation is that in an exponential equation, the variable is in the exponent, while in a logarithmic equation, the variable is in the base. To solve a logarithmic equation, you can rewrite it as an exponential equation and then use the techniques mentioned in the previous question.

How can I check my solution to an exponential equation?

You can check your solution by plugging it back into the original equation and simplifying both sides. If the resulting values are equal, then your solution is correct. You can also use a graphing calculator to graph both sides of the equation and see if the graphs intersect at the solution.

Are there any special cases when solving exponential equations?

Yes, there are a few special cases when solving exponential equations. These include when the base of the exponential term is equal to 1, when the base is equal to 0, and when the base is a negative number. In these cases, different rules and techniques may need to be applied to solve the equation.

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