If ` log_(a)b=2, log_(b)c=2, and log_(3) c= 3 + log_(3)` a,then the value of c/(ab)is ________.

Explanation

To solve the problem, we need to find the value of \( \frac{c}{ab} \) given the logarithmic relationships. Let's break down the solution step by step.

### Step 1: Use the given logarithmic equations
We have:
1. \( \log_a b = 2 \)
2. \( \log_b c = 2 \)
3. \( \log_3 c = 3 + \log_3 a \)

### Step 2: Rewrite the logarithmic equations
From \( \log_a b = 2 \), we can rewrite it in exponential form:
\[ b = a^2 \]

From \( \log_b c = 2 \), we can also rewrite it:
\[ c = b^2 \]

### Step 3: Substitute \( b \) into the equation for \( c \)
Now, substituting \( b = a^2 \) into \( c = b^2 \):
\[ c = (a^2)^2 = a^4 \]

### Step 4: Use the third logarithmic equation
Now we use the third equation \( \log_3 c = 3 + \log_3 a \):
Substituting \( c = a^4 \):
\[ \log_3 (a^4) = 3 + \log_3 a \]
Using the property of logarithms \( \log_b (m^n) = n \cdot \log_b m \):
\[ 4 \log_3 a = 3 + \log_3 a \]

### Step 5: Solve for \( \log_3 a \)
Rearranging the equation:
\[ 4 \log_3 a - \log_3 a = 3 \]
\[ 3 \log_3 a = 3 \]
\[ \log_3 a = 1 \]

### Step 6: Find \( a \)
Converting back from logarithmic form:
\[ a = 3^1 = 3 \]

### Step 7: Find \( b \) and \( c \)
Now substitute \( a = 3 \) back to find \( b \) and \( c \):
\[ b = a^2 = 3^2 = 9 \]
\[ c = a^4 = 3^4 = 81 \]

### Step 8: Calculate \( \frac{c}{ab} \)
Now we can find \( \frac{c}{ab} \):
\[ ab = 3 \cdot 9 = 27 \]
\[ \frac{c}{ab} = \frac{81}{27} = 3 \]

### Final Answer
The value of \( \frac{c}{ab} \) is \( \boxed{3} \).