If `(3)/(2) log a + (2)/(3) log b - 1 = 0`, find the value of `a^(9).b^(4)`.

Answer

To solve the equation \(\frac{3}{2} \log a + \frac{2}{3} \log b - 1 = 0\) and find the value of \(a^9 \cdot b^4\), we can follow these steps:

### Step 1: Rearranging the Equation
Start by isolating the logarithmic terms:
\[ \frac{3}{2} \log a + \frac{2}{3} \log b = 1 \]

### Step 2: Expressing 1 as a Logarithm
We can express 1 in terms of logarithms. Since \(\log_{10} 10 = 1\), we can rewrite the equation as:
\[ \frac{3}{2} \log a + \frac{2}{3} \log b = \log_{10} 10 \]

### Step 3: Applying Logarithmic Properties
Using the property of logarithms that states \(x \log a = \log a^x\), we can rewrite the equation:
\[ \log a^{\frac{3}{2}} + \log b^{\frac{2}{3}} = \log_{10} 10 \]

### Step 4: Combining Logarithms
Using the property \(\log a + \log b = \log(ab)\), we can combine the logarithms:
\[ \log \left( a^{\frac{3}{2}} \cdot b^{\frac{2}{3}} \right) = \log_{10} 10 \]

### Step 5: Removing the Logarithm
Since the logarithm is equal, we can remove it by exponentiating both sides:
\[ a^{\frac{3}{2}} \cdot b^{\frac{2}{3}} = 10 \]

### Step 6: Finding \(a^9 \cdot b^4\)
To find \(a^9 \cdot b^4\), we can manipulate the equation. We can raise both sides to the power of 6:
\[ \left( a^{\frac{3}{2}} \cdot b^{\frac{2}{3}} \right)^6 = 10^6 \]
This simplifies to:
\[ a^{9} \cdot b^{4} = 10^6 \]

### Final Answer
Thus, the value of \(a^9 \cdot b^4\) is:
\[ \boxed{10^6} \]