https://file.aiquickdraw.com/mathbot/file/1726859798271fdk7k4vu.jpg

The equation shown in the image is:

\[
\log_x 1000 - \log_x x^2 + (10^{-3})^0 = 0
\]

Let's break it down step by step:

1. **Simplify \( (10^{-3})^0 \)**:
   \[
   (10^{-3})^0 = 1
   \]
   So, the equation becomes:
   \[
   \log_x 1000 - \log_x x^2 + 1 = 0
   \]

2. **Apply the logarithmic property**: 
   The property of logarithms, \( \log_x a - \log_x b = \log_x \left( \frac{a}{b} \right) \), allows us to combine the first two terms:
   \[
   \log_x \left( \frac{1000}{x^2} \right) + 1 = 0
   \]

3. **Isolate the logarithmic term**:
   Subtract 1 from both sides:
   \[
   \log_x \left( \frac{1000}{x^2} \right) = -1
   \]

4. **Rewrite the logarithmic equation in exponential form**:
   The equation \( \log_x (y) = z \) is equivalent to \( y = x^z \). Thus:
   \[
   \frac{1000}{x^2} = x^{-1}
   \]

5. **Solve for \( x \)**:
   Multiply both sides by \( x^2 \):
   \[
   1000 = x^1
   \]
   So, \( x = 1000 \).

Thus, the solution is \( x = 1000 \).

---

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### Related Questions:
1. How do logarithmic properties simplify equations?
2. What is the general form for converting logarithmic equations to exponential form?
3. How can the properties of exponents be applied to logarithmic equations?
4. What is the significance of logarithmic base changes in equations?
5. How do you handle equations with multiple logarithmic terms?

### Tip:
When dealing with logarithmic equations, always check for simplifications using the properties of logarithms to reduce the complexity of the problem.

Learn how to solve the logarithmic equation log_x 1000 - log_x x^2 + (10^{-3})^0 = 0 with a step-by-step solution. This problem covers key logarithmic concepts, such as applying the properties of logarithms and converting logarithms to exponential form. Suitable for students in Grades 9-12.

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