Question
Consider the following equation. \[ \frac{12}{1+e^{-x}}=2 \] (a) Find the exact solution of the exponential equation in terms of logarithms. \( x= \) (b) Use a calculator to find an approximation to the solution, rounded to six decimal places. \( x=\square \)
Ask by Hobbs Bowers. in the United States
Feb 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x = -\ln(5) \)
Approximately, \( x \approx -1.609438 \)
Solution
We start with the equation
12/(1 + e^(–x)) = 2.
Step 1. Multiply both sides by (1 + e^(–x)) to eliminate the fraction:
12 = 2(1 + e^(–x)).
Step 2. Divide both sides by 2:
6 = 1 + e^(–x).
Step 3. Subtract 1 from both sides to isolate the exponential term:
e^(–x) = 5.
Step 4. Take the natural logarithm of both sides:
ln(e^(–x)) = ln(5).
Since ln(e^(–x)) simplifies to –x, this gives:
–x = ln(5).
Step 5. Multiply both sides by –1:
x = –ln(5).
Thus, the exact solution is: x = –ln(5).
For the approximate value, compute ln(5) ≈ 1.609437912. Therefore:
x ≈ –1.609438 (rounded to six decimal places).
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Simplify this solution Mind Expander
To solve the equation \(\frac{12}{1+e^{-x}}=2\), let's start by isolating the exponential term. First, multiply both sides by \(1 + e^{-x}\): \[ 12 = 2(1 + e^{-x}) \] Expanding the right side gives: \[ 12 = 2 + 2e^{-x} \] Now, subtract \(2\) from both sides: \[ 10 = 2e^{-x} \] Next, divide both sides by \(2\): \[ 5 = e^{-x} \] Taking the natural logarithm of both sides leads to: \[ -x = \ln(5) \] Therefore, we can express \(x\) as: \[ x = -\ln(5) \] Now, for the approximation, using a calculator, we can evaluate \( -\ln(5) \): \( x \approx -1.609438 \) (to six decimal places). So the answers are: (a) \( x = -\ln(5) \) (b) \( x = -1.609439 \)
