We are solving the equation:

\[
0.5 = 1 - e^{-\frac{x}{100}}
\]

### Step 1: Isolate the exponential term.

\[
e^{-\frac{x}{100}} = 1 - 0.5 = 0.5
\]

### Step 2: Take the natural logarithm (ln) of both sides.

\[
\ln \left( e^{-\frac{x}{100}} \right) = \ln(0.5)
\]

Using the property of logarithms, \(\ln(e^y) = y\), we get:

\[
-\frac{x}{100} = \ln(0.5)
\]

### Step 3: Solve for \(x\).

\[
x = -100 \cdot \ln(0.5)
\]

Now let's calculate the value.

\[
x \approx -100 \cdot (-0.6931) = 69.31
\]

Thus, the solution is:

\[
x \approx 69.31
\]

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. What is the significance of the natural logarithm in solving exponential equations?
2. How would the result change if the coefficient in the exponent was different (e.g., \(-x/200\))?
3. Can you express the solution in terms of other logarithmic bases?
4. How would you solve this equation graphically?
5. What are the real-world applications of such exponential decay equations?

**Tip:** Whenever solving exponential equations, isolating the exponential term and using logarithms is key.

Discover how to solve the exponential equation 0.5 = 1 - e^(-x/100) using natural logarithms. This step-by-step guide explores key concepts such as isolating the exponential term and applying logarithmic properties, ideal for students in Grades 10-12.

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