Given log x = 2m - n, log y = n - 2m and log z = 3m - 2n, find in terms of m and n, the value of `"log" (x^(2)y^(3))/(z^(4))`.

Answer

To solve the problem, we need to find the value of \( \log \left( \frac{x^2 y^3}{z^4} \right) \) in terms of \( m \) and \( n \) using the given logarithmic expressions for \( x \), \( y \), and \( z \).

### Step-by-Step Solution:

1. **Use the properties of logarithms**:
We know that:
\[ \log \left( \frac{a}{b} \right) = \log a - \log b \]
and
\[ \log (a^b) = b \cdot \log a \]
Therefore, we can express \( \log \left( \frac{x^2 y^3}{z^4} \right) \) as:
\[ \log \left( \frac{x^2 y^3}{z^4} \right) = \log (x^2) + \log (y^3) - \log (z^4) \]

2. **Apply the power rule**:
Using the power rule, we rewrite the expression:
\[ \log (x^2) = 2 \log x, \quad \log (y^3) = 3 \log y, \quad \log (z^4) = 4 \log z \]
Thus, we have:
\[ \log \left( \frac{x^2 y^3}{z^4} \right) = 2 \log x + 3 \log y - 4 \log z \]

3. **Substitute the given values**:
We substitute the given values of \( \log x \), \( \log y \), and \( \log z \):
- \( \log x = 2m - n \)
- \( \log y = n - 2m \)
- \( \log z = 3m - 2n \)

Therefore:
\[ \log \left( \frac{x^2 y^3}{z^4} \right) = 2(2m - n) + 3(n - 2m) - 4(3m - 2n) \]

4. **Expand and simplify**:
Now, we expand each term:
\[ = 4m - 2n + 3n - 6m - 12m + 8n \]
Combine like terms:
- For \( m \): \( 4m - 6m - 12m = -14m \)
- For \( n \): \( -2n + 3n + 8n = 9n \)

Thus, we have:
\[ \log \left( \frac{x^2 y^3}{z^4} \right) = -14m + 9n \]

### Final Answer:
The value of \( \log \left( \frac{x^2 y^3}{z^4} \right) \) in terms of \( m \) and \( n \) is:
\[ \log \left( \frac{x^2 y^3}{z^4} \right) = -14m + 9n \]