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Let's break down the equation step-by-step:

The given equation is:

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

### Step 1: Simplify the expression \((10^{-3})^0\)
Any number raised to the power of 0 is equal to 1. Therefore, this simplifies to:

\[
(10^{-3})^0 = 1
\]

So, the equation becomes:

\[
\log_x 1000 - \log_x x^2 + 1 = 0
\]

### Step 2: Apply logarithmic properties
Using the property \(\log_x x^a = a\), we can simplify \(\log_x x^2\):

\[
\log_x x^2 = 2
\]

So the equation becomes:

\[
\log_x 1000 - 2 + 1 = 0
\]

### Step 3: Combine constants
\(-2 + 1 = -1\), so:

\[
\log_x 1000 - 1 = 0
\]

### Step 4: Solve for \(\log_x 1000\)
Now, move \(-1\) to the other side:

\[
\log_x 1000 = 1
\]

This implies:

\[
x^1 = 1000
\]

### Step 5: Solve for \(x\)
So, \(x = 1000\).

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

---

Do you want more details or have any questions?

Here are 5 related questions to consider:
1. How do you solve logarithmic equations involving different bases?
2. Can logarithms be used to solve exponential equations?
3. What are the properties of logarithms that help simplify equations like this?
4. How do logarithms apply in real-world problems like sound intensity or earthquake measurements?
5. What happens if the base of a logarithm is less than 1?

**Tip:** Always remember that \(\log_x x^a = a\) is a key property to simplify logarithmic expressions.

Learn how to solve the logarithmic equation log_x 1000 - log_x x^2 + (10^{-3})^0 = 0 using properties of logarithms and exponents. The solution is detailed step by step, including the key concepts of logarithmic properties and simplifying expressions.

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