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Chapter 3: Problem 59

Solve the exponential equation algebraically. Approximate the result to three decimal places. $$e^{2 x}-4 e^{x}-5=0$$

Short Answer

Expert verified
The solution of the equation \(e^{2x} - 4e^x -5 = 0\) is approximately \(x = 1.609\).

Step by step solution

01

Rewrite the equation using substitution

Let \(y = e^x\). This allows us to rewrite the equation in a more simple quadratic form, i.e. \(y^2 - 4y - 5 = 0\).
02

Factor the quadratic equation

The quadratic equation can be factored into \((y - 5)(y + 1) = 0\).
03

Solve for y

Setting each factor equal to zero gives the solutions for y, \(y = 5\) and \(y = -1\).
04

Substitute y back into the equation

Remembering that \(y = e^x\), replace y in the solutions obtained in step 3. Thus, we have \(e^x = 5\) and \(e^x = -1\).
05

Solve for x

To get x, take the natural logarithm of both sides of the equation so we solve for \(x = \ln{5}\) and \(x = \ln{-1}\). We disregard \(x = \ln{-1}\) since it is undefined. Hence, our solution is \(x = \ln{5}\).
06

Approximate x

Using a calculator, the value of \(\ln{5}\) is approximately 1.609, which is the solution of x when approximated to three decimal places.

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Most popular questions from this chapter

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