To solve the given equation:

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

Step-by-step Solution:

1. Simplify the constants:

   • $(10^{-3})^0 = 1$, because any nonzero number raised to the power of 0 equals 1.

   The equation simplifies to: $\log_x 1000 - \log_x x^2 + 1 = 0$

2. Apply the logarithmic property $\log_x x^2 = 2$: $\log_x 1000 - 2 + 1 = 0$

   Simplify further: $\log_x 1000 - 1 = 0$

3. Isolate the logarithm term: $\log_x 1000 = 1$

4. Rewrite the logarithmic equation in exponential form: By the definition of logarithms, $\log_x 1000 = 1$ implies: $x^1 = 1000$

5. Solve for $x$: $x = 1000$

Final Answer:

$x = 1000$

Would you like a detailed explanation of any step? Let me know!

Related Questions:

1. What is the general formula for solving logarithmic equations?
2. How do you simplify expressions with multiple logarithmic terms?
3. What are the properties of logarithms that can help in solving equations?
4. Can you explain the relationship between logarithms and exponential equations?
5. How does the base of the logarithm affect the solution?

Tip:

When solving logarithmic equations, always check that the solution is valid in the original equation (logarithms are undefined for negative or zero arguments).