https://file.aiquickdraw.com/mathbot/file/1730216504093sg0mr749.jpg

The equation shown in the image is:

\[
\frac{82}{1 + e^{-x}} = 4
\]

To solve for \( x \), we can proceed with the following steps:

### Step-by-Step Solution

1. **Isolate the fraction**:
   \[
   \frac{82}{1 + e^{-x}} = 4
   \]

2. **Multiply both sides by \( 1 + e^{-x} \)** to eliminate the denominator:
   \[
   82 = 4(1 + e^{-x})
   \]

3. **Expand the right side**:
   \[
   82 = 4 + 4e^{-x}
   \]

4. **Subtract 4 from both sides**:
   \[
   78 = 4e^{-x}
   \]

5. **Divide by 4** to isolate \( e^{-x} \):
   \[
   e^{-x} = \frac{78}{4} = 19.5
   \]

6. **Take the natural logarithm of both sides** to solve for \( -x \):
   \[
   -x = \ln(19.5)
   \]

7. **Solve for \( x \)** by multiplying both sides by \(-1\):
   \[
   x = -\ln(19.5)
   \]

Using a calculator, we can find the approximate value of \( x \):
\[
x \approx -\ln(19.5) \approx -2.97
\]

### Final Answer
\[
x \approx -2.97
\]

Would you like more details on any step or have any further questions?

### Related Questions
1. How would you solve a similar equation with a different constant in the numerator?
2. What if the constant on the right side was different? How would that affect the solution?
3. How do exponential functions impact the solution of an equation like this?
4. Can this equation be solved graphically? If so, what would the graph look like?
5. What applications in real life might involve equations like this?

### Tip
When dealing with exponential equations, taking the natural logarithm can simplify the process of solving for the exponent.

Solve the equation: 82 / (1 + e^(-x)) = 4

Learn how to solve the exponential equation 82 / (1 + e^(-x)) = 4 with a step-by-step solution. This problem covers key concepts in algebra, exponential functions, and logarithmic properties, suitable for students in Grades 10-12.

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