https://file.aiquickdraw.com/mathbot/file/1729218022956egfb3ztp.jpg

Let's break down the two equations shown in the image and explain why one has a solution and the other does not.

### First Equation:
\[
\log \left( \frac{x-3}{x} \right) = 2
\]

To solve this equation, we can follow these steps:

1. **Exponentiate both sides**:  
   \[
   \frac{x-3}{x} = 10^2 = 100
   \]
   
2. **Solve for \( x \)**:
   \[
   \frac{x-3}{x} = 100
   \]
   Multiply both sides by \( x \):
   \[
   x - 3 = 100x
   \]
   Simplify and solve for \( x \):
   \[
   -3 = 99x \quad \Rightarrow \quad x = -\frac{3}{99} = -\frac{1}{33}
   \]

Now, the solution is \( x = -\frac{1}{33} \). However, **logarithmic functions have a domain restriction**: the argument of the logarithm must be positive. The expression inside the logarithm is \( \frac{x-3}{x} \). For \( x = -\frac{1}{33} \), the argument becomes negative, so **this value of \( x \) is not valid**. However, the setup of the equation suggests that there is an initial step where the logarithm could have a potential solution before testing the domain.

### Second Equation:
\[
\log(x - 3) - \log(x - 2) = 2
\]

1. **Combine the logarithms using properties**:
   \[
   \log \left( \frac{x - 3}{x - 2} \right) = 2
   \]
   
2. **Exponentiate both sides**:
   \[
   \frac{x - 3}{x - 2} = 10^2 = 100
   \]

3. **Solve for \( x \)**:
   \[
   x - 3 = 100(x - 2)
   \]
   Expand and simplify:
   \[
   x - 3 = 100x - 200 \quad \Rightarrow \quad -3 + 200 = 100x - x \quad \Rightarrow \quad 197 = 99x \quad \Rightarrow \quad x = \frac{197}{99}
   \]

Now, let's check the domain:
- For \( \log(x-3) \), the argument must be positive, meaning \( x > 3 \).
- For \( \log(x-2) \), the argument must also be positive, meaning \( x > 2 \).

The solution \( x = \frac{197}{99} \approx 1.99 \) does **not** satisfy \( x > 3 \), so there is **no valid solution** to this equation.

### Summary:
- The first equation **has a solution** because the log expression can work within the domain, even though the solution itself is invalid for a real \( x \).
- The second equation **has no solution** because the domain restrictions make it impossible to satisfy the inequality \( x > 3 \).

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

Here are 5 related questions:
1. How can we expand on logarithmic properties to combine more complex expressions?
2. What are the general domain restrictions of logarithmic functions?
3. Can logarithmic equations ever yield negative solutions?
4. How do we interpret extraneous solutions in logarithmic equations?
5. What is the geometric meaning of logarithms in relation to these solutions?

**Tip**: Always check the domain of the logarithmic function before finalizing a solution!

Explain why the equation log((x-3)/x) = 2 has a solution but the equation log(x - 3) - log(x - 2) = 2 has no solutions.

This problem explains why the equation log((x-3)/x) = 2 has a solution while the equation log(x - 3) - log(x - 2) = 2 does not. It covers logarithmic properties, domain restrictions, and the steps to solve both equations. Suitable for students in Grades 10-12 studying Precalculus.

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