\(\log _{(x)}(1000)-\log \left(x^{(2)}\right)=-1\)

We are given the equation:

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

### Step 1: Use logarithmic properties
We can use logarithmic properties to simplify the equation. First, apply the power rule of logarithms, which states that:

\[
\log_b(a^n) = n \log_b(a)
\]

Thus, \(\log(x^2)\) becomes:

\[
\log(x^2) = 2 \log(x)
\]

Now, substitute this into the original equation:

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

### Step 2: Convert to a common logarithmic base
Next, we need to express the logarithms with a common base. Using the change of base formula, \(\log_b(a) = \frac{\log(a)}{\log(b)}\), convert \(\log_x(1000)\) to base 10 logarithms:

\[
\log_x(1000) = \frac{\log(1000)}{\log(x)}
\]

Since \(\log(1000) = 3\) (because \(1000 = 10^3\)), this becomes:

\[
\log_x(1000) = \frac{3}{\log(x)}
\]

Substitute this into the equation:

\[
\frac{3}{\log(x)} - 2 \log(x) = -1
\]

### Step 3: Solve the equation
Now, let's solve for \(x\). Multiply both sides by \(\log(x)\) to eliminate the fraction:

\[
3 - 2 \log(x) \cdot \log(x) = - \log(x)
\]

This simplifies to:

\[
3 - 2 (\log(x))^2 = - \log(x)
\]

Rearrange the terms:

\[
2 (\log(x))^2 - \log(x) - 3 = 0
\]

### Step 4: Solve the quadratic equation
This is a quadratic equation in terms of \(\log(x)\). Let \(y = \log(x)\), then the equation becomes:

\[
2y^2 - y - 3 = 0
\]

We can solve this quadratic equation using the quadratic formula:

\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \(2y^2 - y - 3 = 0\), we have:
- \(a = 2\),
- \(b = -1\),
- \(c = -3\).

Substitute these values into the quadratic formula:

\[
y = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(2)(-3)}}{2(2)}
\]

Simplify:

\[
y = \frac{1 \pm \sqrt{1 + 24}}{4} = \frac{1 \pm \sqrt{25}}{4}
\]

\[
y = \frac{1 \pm 5}{4}
\]

Thus, we have two possible solutions for \(y\):

\[
y = \frac{1 + 5}{4} = \frac{6}{4} = 1.5 \quad \text{or} \quad y = \frac{1 - 5}{4} = \frac{-4}{4} = -1
\]

### Step 5: Find \(x\)
Recall that \(y = \log(x)\), so we can now find \(x\) for each solution.

1. When \(y = 1.5\), we have \(\log(x) = 1.5\), so:

\[
x = 10^{1.5} = \sqrt{10^3} = \sqrt{1000} \approx 31.62
\]

2. When \(y = -1\), we have \(\log(x) = -1\), so:

\[
x = 10^{-1} = 0.1
\]

### Final Answer:
Thus, the two possible values for \(x\) are approximately:

\[
x \approx 31.62 \quad \text{or} \quad x = 0.1
\]

Would you like further details or clarification on any part of the solution? 

### Related questions:
1. How do you derive the change of base formula for logarithms?
2. What is the meaning of the logarithmic property \(\log_b(a^n) = n \log_b(a)\)?
3. Can the logarithmic equation have a solution where \(x\) is negative?
4. How can you check if the solutions satisfy the original logarithmic equation?
5. What are the key differences between logarithmic and exponential equations?

### Tip:
When solving logarithmic equations, always ensure that the arguments of the logarithms are positive, as the logarithm of a non-positive number is undefined in real numbers.

Discover how to solve the logarithmic equation log_x(1000) - log(x^2) = -1 through a detailed, step-by-step process. Learn about key concepts such as the change of base formula, logarithmic properties, and quadratic equations. This solution is ideal for high school students in Grades 10-12.

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