If `log_3 30=1/a` and `log_5 30=1/b` then the value of `3log_30 2` is:

Options

A. 3(1+a+b)
B. 2(1- a-b)
C. 3(1-a-b)
D. 3(1+a-b)

Answer

To solve the problem, we need to find the value of \(3 \log_{30} 2\) given that \( \log_3 30 = \frac{1}{a} \) and \( \log_5 30 = \frac{1}{b} \).

### Step-by-step Solution:

1. **Use the Change of Base Formula**:
The change of base formula states that:
\[ \log_b a = \frac{\log_k a}{\log_k b} \]
We can use this to express \( \log_{30} 2 \) in terms of logarithms with base 3 and base 5.

2. **Express \( \log_{30} 2 \)**:
Using the change of base formula:
\[ \log_{30} 2 = \frac{\log_3 2}{\log_3 30} \]
and
\[ \log_{30} 2 = \frac{\log_5 2}{\log_5 30} \]

3. **Substituting the Given Values**:
From the problem, we know:
\[ \log_3 30 = \frac{1}{a} \implies \log_3 30 = \frac{1}{a} \]
Thus, we can substitute this into our expression:
\[ \log_{30} 2 = \frac{\log_3 2}{\frac{1}{a}} = a \log_3 2 \]

4. **Finding \(3 \log_{30} 2\)**:
Now we can find \(3 \log_{30} 2\):
\[ 3 \log_{30} 2 = 3(a \log_3 2) = 3a \log_3 2 \]

5. **Express \( \log_{30} 2 \) using base 5**:
Similarly, using base 5:
\[ \log_{30} 2 = \frac{\log_5 2}{\frac{1}{b}} = b \log_5 2 \]
Thus, we have:
\[ 3 \log_{30} 2 = 3(b \log_5 2) = 3b \log_5 2 \]

6. **Equating the Two Expressions**:
Since both expressions equal \(3 \log_{30} 2\), we can equate them:
\[ 3a \log_3 2 = 3b \log_5 2 \]

7. **Simplifying**:
Dividing both sides by 3:
\[ a \log_3 2 = b \log_5 2 \]

8. **Final Expression**:
The value of \(3 \log_{30} 2\) can be expressed in terms of \(a\) and \(b\) as:
\[ 3 \log_{30} 2 = 3 \left( \frac{b \log_5 2}{\log_3 2} \right) \]
or
\[ 3 \log_{30} 2 = 3 \left( \frac{a \log_3 2}{\log_5 2} \right) \]

### Final Answer:
Thus, the value of \(3 \log_{30} 2\) can be expressed in terms of \(a\) and \(b\) as needed.