Let ` log x= 2m - 3n and log y = 3n - 2m ` Find the value of log `(x^(3) /y ^(2)) ` in terms of m and n.

Answer

To find the value of \( \log \left( \frac{x^3}{y^2} \right) \) in terms of \( m \) and \( n \), we will follow these steps:

### Step 1: Write down the given logarithmic expressions
We have:
\[ \log x = 2m - 3n \]
\[ \log y = 3n - 2m \]

### Step 2: Use the logarithmic property for powers
Using the property \( \log a^n = n \log a \), we can find \( \log x^3 \) and \( \log y^2 \).

For \( \log x^3 \):
\[ \log x^3 = 3 \log x = 3(2m - 3n) = 6m - 9n \]

For \( \log y^2 \):
\[ \log y^2 = 2 \log y = 2(3n - 2m) = 6n - 4m \]

### Step 3: Use the logarithmic property for division
Now we can use the property \( \log \left( \frac{a}{b} \right) = \log a - \log b \):
\[ \log \left( \frac{x^3}{y^2} \right) = \log x^3 - \log y^2 \]

### Step 4: Substitute the values we found
Substituting the values we calculated:
\[ \log \left( \frac{x^3}{y^2} \right) = (6m - 9n) - (6n - 4m) \]

### Step 5: Simplify the expression
Now simplify the expression:
\[ \log \left( \frac{x^3}{y^2} \right) = 6m - 9n - 6n + 4m \]
Combine like terms:
\[ = (6m + 4m) + (-9n - 6n) = 10m - 15n \]

### Final Answer
Thus, the value of \( \log \left( \frac{x^3}{y^2} \right) \) in terms of \( m \) and \( n \) is:
\[ \log \left( \frac{x^3}{y^2} \right) = 10m - 15n \]