If ` log_(10) x = y ," then find "log_(1000)x^(2)" in terms of " y`.

Explanation

To solve the problem, we need to find \( \log_{1000} x^2 \) in terms of \( y \), given that \( \log_{10} x = y \).

### Step-by-Step Solution:

1. **Express \( \log_{1000} x^2 \) using the change of base formula:**
\[ \log_{1000} x^2 = \frac{\log_{10} x^2}{\log_{10} 1000} \]

**Hint:** Remember that the change of base formula states that \( \log_a b = \frac{\log_c b}{\log_c a} \).

2. **Simplify \( \log_{10} x^2 \) using the power property of logarithms:**
\[ \log_{10} x^2 = 2 \log_{10} x \]

**Hint:** The property \( \log_a (b^n) = n \log_a b \) is useful here.

3. **Substitute \( \log_{10} x \) with \( y \):**
\[ \log_{10} x^2 = 2y \]

**Hint:** Since \( \log_{10} x = y \), you can replace \( \log_{10} x \) with \( y \).

4. **Calculate \( \log_{10} 1000 \):**
\[ \log_{10} 1000 = \log_{10} (10^3) = 3 \]

**Hint:** Remember that \( 1000 = 10^3 \), so you can use the property of logarithms for powers.

5. **Substitute back into the expression for \( \log_{1000} x^2 \):**
\[ \log_{1000} x^2 = \frac{2y}{3} \]

**Hint:** After substituting \( \log_{10} x^2 \) and \( \log_{10} 1000 \), simplify the fraction.

### Final Answer:
\[ \log_{1000} x^2 = \frac{2y}{3} \]