Given : `(log x)/(log y) = (3)/(2)` and log(xy) = 5, find the values of x and y.

Answer

To solve the problem step by step, we will follow the given equations and manipulate them to find the values of \( x \) and \( y \).

### Step 1: Start with the given equations
We have two equations:
1. \(\frac{\log x}{\log y} = \frac{3}{2}\)
2. \(\log(xy) = 5\)

### Step 2: Rewrite the first equation
From the first equation, we can express \(\log y\) in terms of \(\log x\):
\[ \log y = \frac{2}{3} \log x \]

### Step 3: Use the second equation
The second equation states that:
\[ \log(xy) = \log x + \log y = 5 \]
Substituting the expression for \(\log y\) from Step 2 into this equation gives:
\[ \log x + \frac{2}{3} \log x = 5 \]

### Step 4: Combine the logarithmic terms
Combine the terms on the left side:
\[ \log x + \frac{2}{3} \log x = \frac{3}{3} \log x + \frac{2}{3} \log x = \frac{5}{3} \log x \]
Thus, we have:
\[ \frac{5}{3} \log x = 5 \]

### Step 5: Solve for \(\log x\)
To isolate \(\log x\), multiply both sides by \(\frac{3}{5}\):
\[ \log x = 5 \cdot \frac{3}{5} = 3 \]

### Step 6: Find \( x \)
Now, we can find \( x \) by converting from logarithmic form:
\[ x = 10^{\log x} = 10^3 = 1000 \]

### Step 7: Find \(\log y\)
Now, substitute \(\log x\) back into the equation for \(\log y\):
\[ \log y = \frac{2}{3} \cdot 3 = 2 \]

### Step 8: Find \( y \)
Convert from logarithmic form to find \( y \):
\[ y = 10^{\log y} = 10^2 = 100 \]

### Final Answer
Thus, the values of \( x \) and \( y \) are:
\[ x = 1000, \quad y = 100 \]