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

Options

A. 1
B. 3
C. 9
D. 27

Explanation

To solve the problem, we will follow the steps outlined in the video transcript and apply logarithmic properties systematically.

### Step-by-Step Solution:

1. **Given Information:**
- We have three equations:
1. \( \log_a b = 2 \)
2. \( \log_b c = 2 \)
3. \( \log_3 c = 3 + \log_3 a \)

2. **Convert the first equation using the change of base formula:**
\[ \log_a b = 2 \implies \frac{\log b}{\log a} = 2 \implies \log b = 2 \log a \]

3. **Substitute \( \log b \) into the second equation:**
\[ \log_b c = 2 \implies \frac{\log c}{\log b} = 2 \implies \log c = 2 \log b \]
Now substituting \( \log b = 2 \log a \):
\[ \log c = 2(2 \log a) = 4 \log a \]

4. **Express \( c \) in terms of \( a \):**
\[ \log c = 4 \log a \implies c = a^4 \]

5. **Use the third equation:**
\[ \log_3 c = 3 + \log_3 a \]
This can be rewritten using the logarithmic quotient rule:
\[ \log_3 c - \log_3 a = 3 \implies \log_3 \left(\frac{c}{a}\right) = 3 \]
Thus,
\[ \frac{c}{a} = 3^3 = 27 \implies c = 27a \]

6. **Equate the two expressions for \( c \):**
From step 4, we have \( c = a^4 \) and from step 5, \( c = 27a \):
\[ a^4 = 27a \]
Dividing both sides by \( a \) (assuming \( a \neq 0 \)):
\[ a^3 = 27 \implies a = 3 \]

7. **Find \( c \):**
Substitute \( a = 3 \) into \( c = a^4 \):
\[ c = 3^4 = 81 \]

8. **Find \( b \):**
Using the first equation \( \log_a b = 2 \):
\[ \log_3 b = 2 \implies b = 3^2 = 9 \]

9. **Calculate \( \frac{c}{ab} \):**
Now we have \( c = 81 \), \( a = 3 \), and \( b = 9 \):
\[ ab = 3 \times 9 = 27 \]
Therefore,
\[ \frac{c}{ab} = \frac{81}{27} = 3 \]

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