Let's solve the equation step by step.

The equation is:

\[
\log{x}(1000) - \log{x^2} = -1
\]

### Step 1: Apply logarithmic properties
Use the power rule of logarithms to simplify \(\log(x^2)\):

\[
\log_x(1000) - 2\log_x(x) = -1
\]

But \(\log_x(x)\) is just 1, so the equation becomes:

\[
\log_x(1000) - 2 = -1
\]

### Step 2: Solve for \(\log_x(1000)\)
Now, add 2 to both sides:

\[
\log_x(1000) = 1
\]

### Step 3: Convert from logarithmic form to exponential form
Rewrite the logarithmic equation in exponential form:

\[
x^1 = 1000
\]

So, \(x = 1000\).

Thus, the solution to the equation is:

\[
x = 1000
\]

Would you like more details on any step?

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Here are some related questions:
1. What are the key logarithmic properties?
2. How do you convert between logarithmic and exponential forms?
3. How do you simplify logarithmic expressions?
4. Can logarithmic equations have multiple solutions?
5. What happens if you change the base of a logarithm?

**Tip:** Always check the domain of the logarithmic function to ensure the argument is positive.

Explore how to solve the logarithmic equation logx(1000) - log(x^2) = -1 step by step. This solution covers key logarithmic properties, including simplifying logarithmic expressions and converting between logarithmic and exponential forms. Suitable for students in Grades 9-12.

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