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Chapter 3: Problem 64

Solve the exponential equation algebraically. Approximate the result to three decimal places. $$\frac{400}{1+e^{-x}}=350$$

Short Answer

Expert verified
The solution to the given exponential equation, approximated to three decimal places, is \(x \approx 1.947\).

Step by step solution

01

Isolating the exponential function

Firstly, let's isolate the exponential function on one side. This can be done by subtracting 350 from both sides of the equation, resulting in: \(\frac{400}{1+e^{-x}} - 350 = 0\). Consequently, this simplifies to \(\frac{400}{1+e^{-x}} = 350\), and after multiplying by \(1+e^{-x}\), it gives \(400 = 350(1+e^{-x})\).
02

Solving for \(e^{-x}\)

Now, solve for \(e^{-x}\). Divide both sides by 350, which results in \(\frac{400}{350} = 1+e^{-x}\). After rearranging the terms, you get \(e^{-x} = \frac{400}{350} - 1\). If you simplify that, you obtain \(e^{-x} = \frac{50}{350}\) or \(e^{-x} = \frac{1}{7}\).
03

Applying natural logarithm

Apply the natural logarithm (ln) to both sides of the equation to isolate \(x\). This results in \(-x= ln(\frac{1}{7})\). Now, multiply both sides by -1 so that you are left with \(x = -ln(\frac{1}{7})\).
04

Approximating the result

To find the approximate value of \(x\), just substitute \(\frac{1}{7}\) into the natural logarithm and multiply by -1. This will give you the decimal representation of \(x\). After evaluating it, you get \(x \approx 1.947\).

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