To solve the equation \( \log(1 - x) = 0.5 \), follow these steps:

### Step 1: Convert the logarithmic equation to its exponential form.

The equation \( \log(1 - x) = 0.5 \) is in the logarithmic form. By definition, if \( \log_b(y) = x \), then \( y = b^x \). Here, we assume that the base of the logarithm is 10 (common logarithm). Thus, the equation becomes:

\[
1 - x = 10^{0.5}
\]

### Step 2: Simplify the exponential.

Calculate \( 10^{0.5} \), which is the square root of 10:

\[
1 - x = \sqrt{10}
\]

\[
1 - x = 3.162
\]

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

Now, solve for \( x \) by subtracting 1 from both sides:

\[
-x = 3.162 - 1
\]

\[
-x = 2.162
\]

Multiply both sides by -1:

\[
x = -2.162
\]

### Final Answer:
The solution to the equation \( \log(1 - x) = 0.5 \) is:

\[
x = -2.162
\]

Would you like more details or have any questions?

Here are some related questions:

1. How can logarithmic functions be converted to exponential form?
2. What are the common and natural logarithms, and how do they differ?
3. How do you solve logarithmic equations with different bases?
4. How does the domain of the logarithmic function affect solutions to equations like this?
5. What are some real-world applications of logarithmic functions?

**Tip:** When solving logarithmic equations, always check if the argument of the logarithm remains positive for the solution found, as the logarithm is undefined for non-positive numbers.

Learn how to solve the logarithmic equation log(1 - x) = 0.5. This step-by-step guide converts the logarithmic form to an exponential equation, simplifies the expression, and finds the solution. Suitable for high school students in Grades 10-12, covering logarithmic and exponential concepts.

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