https://file.aiquickdraw.com/mathbot/file/1726904978336ncni4v9r.jpg

The equation provided in the image is:

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

Let’s solve this step by step:

### Step 1: Simplify the exponent and constant terms
\[
(10^{-3})^0 = 1
\]
because any non-zero number raised to the power of 0 is 1.

This simplifies the equation to:
\[
\log_x 1000 - \log_x x^2 + 1 = 0
\]

### Step 2: Apply logarithmic properties
Using the logarithmic property:
\[
\log_x a - \log_x b = \log_x \left(\frac{a}{b}\right)
\]

We can simplify the equation:
\[
\log_x \left(\frac{1000}{x^2}\right) + 1 = 0
\]

### Step 3: Move the constant term
Now, subtract 1 from both sides:
\[
\log_x \left(\frac{1000}{x^2}\right) = -1
\]

### Step 4: Convert the logarithmic equation to exponential form
Using the fact that:
\[
\log_x A = B \implies A = x^B
\]

We can rewrite the equation as:
\[
\frac{1000}{x^2} = x^{-1}
\]

### Step 5: Simplify the equation
Multiply both sides of the equation by \(x^2\) to eliminate the fraction:
\[
1000 = x
\]

### Step 6: Final solution
Thus, the value of \(x\) is:
\[
x = 1000
\]

### Final Answer:
The solution to the equation is \(x = 1000\).

Would you like further clarification or a detailed breakdown? 

Here are some related questions to consider:
1. What are the basic logarithmic properties?
2. How do you convert a logarithmic equation to its exponential form?
3. What are the rules for exponents, particularly when simplifying expressions like \((10^{-3})^0\)?
4. Can logarithms with different bases be solved using the change-of-base formula?
5. How do logarithmic scales relate to real-world problems, like in sound intensity or earthquake measurements?

**Tip:** When solving logarithmic equations, always look for ways to apply logarithmic properties to simplify terms, like combining or expanding logs.

Learn how to solve the logarithmic equation log_x 1000 - log_x x^2 + (10^-3)^0 = 0 step by step. This problem covers logarithmic properties, exponent rules, and algebraic techniques. Suitable for students in Grades 9-12.

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