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.