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Chapter 3: Problem 63

Solve the exponential equation algebraically. Approximate the result to three decimal places. $$\frac{500}{100-e^{x / 2}}=20$$

Short Answer

Expert verified
The approximate solution to the equation is \( x = 8.635 \) to three decimal places.

Step by step solution

01

Remove the fraction

To simplify working with the equation, aim to get rid of the denominator. This can be done by multiplying each side of the equation by the denominator: \( \frac{500}{100-e^{x / 2}} \times (100 - e^{x / 2}) = 20 \times (100 - e^{x / 2}) \). This yields: \( 500 = 2000 - 20 \times e^{x / 2} \).
02

Isolate the exponential term

Rearrange the equation to isolate the exponential term \( e^{x / 2} \) on one side of the equation: \( 20 \times e^{x / 2} = 2000 - 500 \) which simplifies to: \( 20 \times e^{x / 2} = 1500 \) and further simplifies to \( e^{x / 2} = 75 \).
03

Apply natural logarithm

To eliminate the exponential, apply a natural logarithm on both sides of the equation. This gives: \( \ln(e^{x / 2}) = \ln(75) \) which simplifies to \( \frac{x}{2} \ln(e) = \ln(75) \). Since \( \ln(e) = 1 \), this further simplifies to \( \frac{x}{2} = \ln(75) \).
04

Solve for x

Multiply the equation by 2 in order to solve for the variable x: \( x = 2 \times \ln(75) \).
05

Numerical approximation

If we substitute the value of \( \ln(75) \) which is approximately 4.3175 into the equation we get \( x=2 \times 4.3175 \), thus the approximate value of \(x\) is 8.635

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