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determine all solutions of x of the equation

To solve the equation \(\log_x(-x + 2) = 2\), we can follow these steps:

### Step 1: Rewrite the logarithmic equation in exponential form
The given equation is:

\[
\log_x(-x + 2) = 2
\]

This can be rewritten in exponential form as:

\[
-x + 2 = x^2
\]

### Step 2: Rearrange the equation to standard quadratic form
Next, let's move all the terms to one side of the equation:

\[
x^2 + x - 2 = 0
\]

### Step 3: Solve the quadratic equation
We can solve the quadratic equation \(x^2 + x - 2 = 0\) using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \(a = 1\), \(b = 1\), and \(c = -2\). Plugging in these values:

\[
x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-2)}}{2(1)}
\]

\[
x = \frac{-1 \pm \sqrt{1 + 8}}{2}
\]

\[
x = \frac{-1 \pm \sqrt{9}}{2}
\]

\[
x = \frac{-1 \pm 3}{2}
\]

### Step 4: Find the roots
This gives us two potential solutions for \(x\):

1. \(x = \frac{-1 + 3}{2} = \frac{2}{2} = 1\)
2. \(x = \frac{-1 - 3}{2} = \frac{-4}{2} = -2\)

### Step 5: Verify the solutions
To ensure these solutions are valid, we need to check that they satisfy the original equation.

- **For \(x = 1\):**

\[
\log_1(-1 + 2) = \log_1(1)
\]

This is undefined because the base \(1\) is not valid in logarithms.

- **For \(x = -2\):**

\[
\log_{-2}(-(-2) + 2) = \log_{-2}(4)
\]

This is also not valid because the logarithm of a negative base is undefined.

### Conclusion
Both potential solutions \(x = 1\) and \(x = -2\) do not satisfy the original equation due to the invalid logarithmic conditions. Therefore, **there are no solutions** to this equation.

Would you like more details or have any questions?

### Related Questions
1. What are the conditions for the existence of a logarithm?
2. How do you solve logarithmic equations with different bases?
3. What are the properties of logarithms that can help in solving equations?
4. How can the quadratic formula be derived?
5. What are the common mistakes to avoid when solving logarithmic equations?

### Tip
Always check the domain of the logarithmic function when solving such equations to ensure the solutions are valid.

Learn how to solve the logarithmic equation log_x(-x + 2) = 2 with detailed steps and verification of solutions. This problem involves logarithmic functions and quadratic equations. Suitable for high school students.

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