https://file.aiquickdraw.com/mathbot/file/1725771970344as3ba3hg.jpg

The equation given in the image is:

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

Let's break this down:

1. \(\log_x 1000\): This is the logarithm of 1000 with base \(x\).
2. \(\log_x x^2\): Using the logarithm property \(\log_b b^a = a\), this simplifies to \(2\), since \(\log_x x^2 = 2\).
3. \((10^{-3})^0\): Any number raised to the power 0 equals 1, so this term equals 1.

Thus, the equation becomes:

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

Simplifying:

\[
\log_x 1000 - 1 = 0
\]

Adding 1 to both sides:

\[
\log_x 1000 = 1
\]

This equation means that \(x\) raised to the power of 1 equals 1000. Therefore, \(x = 1000\).

Thus, the solution to the equation is:

\[
x = 1000
\]

Let me know if you'd like further clarification or details. 

Here are some related questions to explore:

1. What are the properties of logarithms that can simplify complex expressions?
2. How does changing the base of the logarithm affect the solution?
3. Can logarithms with non-positive numbers as bases be defined?
4. How can logarithmic equations be solved if the variable is in both the base and the argument?
5. What is the general formula for converting logarithms from one base to another?

**Tip:** Always remember that the base of a logarithm must be positive and not equal to 1.

Learn how to solve the logarithmic equation log_x 1000 - log_x x^2 + (10^{-3})^0 = 0 with step-by-step explanations. Explore logarithmic properties, exponent rules, and algebraic manipulation. Suitable for students in Grades 10-12.

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