To solve the equation \( e^{-4x} = 2 \), follow these steps:

### Step 1: Take the natural logarithm (ln) of both sides
Since the natural logarithm is the inverse of the exponential function, we can apply \( \ln \) to both sides of the equation:

\[
\ln(e^{-4x}) = \ln(2)
\]

### Step 2: Simplify using the logarithmic identity
We use the logarithmic identity \( \ln(e^y) = y \):

\[
-4x = \ln(2)
\]

### Step 3: Solve for \( x \)
Now, solve for \( x \) by dividing both sides by \(-4\):

\[
x = \frac{\ln(2)}{-4}
\]

### Step 4: Approximate the value of \( x \)
Using \( \ln(2) \approx 0.6931 \):

\[
x = \frac{0.6931}{-4} \approx -0.1733
\]

Thus, the solution is:

\[
x \approx -0.1733
\]

Would you like more details or have any questions?

Here are 5 related questions to further explore this topic:
1. How would the solution change if the equation was \( e^{4x} = 2 \)?
2. What happens when you solve \( e^{-4x} = 1 \)?
3. Can you solve \( e^{-4x} = 5 \) using the same method?
4. How does this technique apply to solving equations like \( a^{bx} = c \) for different values of \( a \) and \( b \)?
5. How can the solution process change for non-base \( e \) exponential functions, such as \( 10^{-4x} = 2 \)?

**Tip:** The natural logarithm \( \ln(x) \) is especially useful for solving equations involving the base \( e \) exponential function.

This guide demonstrates how to solve the exponential equation e^-4x = 2 using logarithms. Learn how to apply natural logarithms to both sides of the equation, use logarithmic identities, and simplify the equation. Includes a step-by-step solution with final answer x ≈ -0.1733. Suitable for high school students in Grades 10-12.

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