solve for x if: logx 1000 - log x^2 + (10^-3)^0 = 0

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.

Learn how to solve the logarithmic equation logx 1000 - log x^2 + (10^-3)^0 = 0 with a detailed step-by-step solution. This problem covers key concepts in logarithms and exponents, using logarithmic properties and algebraic manipulation. Suitable for students in Grades 10-12.

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