If ` log xy^(3) = m and log x ^(3) y ^(2) = p, " find " log (x^(2) -: y) ` in terms of m and p.

Answer

To solve the problem, we need to find \( \log \left( \frac{x^2}{y} \right) \) in terms of \( m \) and \( p \), given that:

1. \( \log (xy^3) = m \)
2. \( \log (x^3y^2) = p \)

### Step 1: Rewrite the expression
We start with the expression we need to find:
\[ \log \left( \frac{x^2}{y} \right) \]
Using the logarithmic property \( \log \left( \frac{a}{b} \right) = \log a - \log b \), we can rewrite this as:
\[ \log (x^2) - \log (y) \]

### Step 2: Express \( \log (x^2) \)
Using the property \( \log (a^b) = b \cdot \log a \), we can express \( \log (x^2) \) as:
\[ \log (x^2) = 2 \log (x) \]
Thus, we have:
\[ \log \left( \frac{x^2}{y} \right) = 2 \log (x) - \log (y) \]

### Step 3: Express \( \log (x) \) and \( \log (y) \)
Next, we need to express \( \log (x) \) and \( \log (y) \) in terms of \( m \) and \( p \).

From the first equation:
\[ \log (xy^3) = \log (x) + \log (y^3) = \log (x) + 3 \log (y) = m \]
This can be rearranged to give:
\[ \log (x) + 3 \log (y) = m \quad \text{(1)} \]

From the second equation:
\[ \log (x^3y^2) = 3 \log (x) + 2 \log (y) = p \]
This can be rearranged to give:
\[ 3 \log (x) + 2 \log (y) = p \quad \text{(2)} \]

### Step 4: Solve the system of equations
Now we have two equations:
1. \( \log (x) + 3 \log (y) = m \)
2. \( 3 \log (x) + 2 \log (y) = p \)

Let \( \log (x) = a \) and \( \log (y) = b \). Then we can rewrite the equations as:
1. \( a + 3b = m \)
2. \( 3a + 2b = p \)

We can solve this system of equations for \( a \) and \( b \).

From equation (1):
\[ a = m - 3b \]

Substituting \( a \) into equation (2):
\[ 3(m - 3b) + 2b = p \]
Expanding this gives:
\[ 3m - 9b + 2b = p \]
Combining like terms:
\[ 3m - 7b = p \]
Rearranging gives:
\[ 7b = 3m - p \]
Thus:
\[ b = \frac{3m - p}{7} \]

Now substituting \( b \) back into the expression for \( a \):
\[ a = m - 3\left(\frac{3m - p}{7}\right) \]
This simplifies to:
\[ a = m - \frac{9m - 3p}{7} = \frac{7m - (9m - 3p)}{7} = \frac{3p - 2m}{7} \]

### Step 5: Substitute \( a \) and \( b \) back
Now we have \( a \) and \( b \):
\[ \log (x) = \frac{3p - 2m}{7}, \quad \log (y) = \frac{3m - p}{7} \]

### Step 6: Substitute into the expression for \( \log \left( \frac{x^2}{y} \right) \)
Now substituting \( a \) and \( b \) back into the expression for \( \log \left( \frac{x^2}{y} \right) \):
\[ \log \left( \frac{x^2}{y} \right) = 2a - b \]
Substituting the values:
\[ = 2\left(\frac{3p - 2m}{7}\right) - \left(\frac{3m - p}{7}\right) \]
This simplifies to:
\[ = \frac{6p - 4m - 3m + p}{7} = \frac{7p - 7m}{7} = p - m \]

### Final Answer
Thus, we find that:
\[ \log \left( \frac{x^2}{y} \right) = p - m \]