We are given the equation:

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

Let's break this down and solve for $x$.

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

Any non-zero number raised to the power of 0 equals 1:

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

Now, the equation becomes:

$\log_x 1000 - \log x^2 + 1 = 0$

Step 2: Rearrange the equation

We can rewrite the equation as:

$\log_x 1000 - \log x^2 = -1$

Step 3: Use logarithmic properties

We can simplify the terms using the following logarithmic properties:

1. $\log_b (a^c) = c \cdot \log_b a$
2. $\log_b a - \log_b c = \log_b \left(\frac{a}{c}\right)$

Apply these properties:

• The term $\log x^2$ becomes $2 \cdot \log x$.

Thus, the equation becomes:

$\log_x 1000 - 2 = -1$

Step 4: Solve for $\log_x 1000$

Now, rearrange the equation:

$\log_x 1000 = -1 + 2$

$\log_x 1000 = 1$

Step 5: Solve for $x$

The equation $\log_x 1000 = 1$ implies that $x^1 = 1000$. Therefore, we have:

$x = 1000$

Final Answer:

The value of $x$ is 1000.

Would you like more details, or do you have any questions?

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Here are 5 related questions to deepen your understanding:

1. How does the logarithmic base change formula work in equations?
2. What are the properties of exponents that are useful for logarithmic simplifications?
3. Can you solve the equation $\log_x 100 - \log x^3 + 2 = 0$?
4. How do we solve logarithmic equations when the base is not given explicitly?
5. What is the relationship between logarithms and exponents?

Tip: Always remember the logarithmic identity $\log_b (a^c) = c \cdot \log_b a$, as it's useful for simplifying many logarithmic expressions.