The given expression is:

$\log_x 1000 - \log_x x^2 + \left(10^{-3}\right)^0$

Let’s break it down step by step:

Step 1: Simplifying $(10^{-3})^0$

Any non-zero number raised to the power of 0 is 1. So:

$(10^{-3})^0 = 1$

Step 2: Simplifying the logarithmic terms

The logarithmic terms are written as follows:

$\log_x 1000 - \log_x x^2$

Using the logarithmic identity $\log_b a - \log_b c = \log_b \left(\frac{a}{c}\right)$, we can combine the logs:

$\log_x \left(\frac{1000}{x^2}\right)$

Now the expression becomes:

$\log_x \left(\frac{1000}{x^2}\right) + 1$

Final Expression:

$\log_x \left(\frac{1000}{x^2}\right) + 1$

This is the simplified form of the original expression.

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Do you want more details or have any questions?

Here are 5 relative questions that build on this explanation:

1. How do you simplify logarithmic expressions with different bases?
2. Can you solve for $x$ if $\log_x 1000 - \log_x x^2 = 2$?
3. What are the properties of exponents and logarithms in simplifying expressions?
4. How does the change of base formula work in logarithms?
5. What are common logarithmic identities and how are they applied?

Tip: Always remember that any number raised to the power of 0 is 1, which is helpful in simplifying expressions involving exponents.