The value of `((1)/(log_(3)12)+(1)/(log_(4)12))` is

Options

A. `2`
B. `0`
C. `1`
D. `2`

Answer

To solve the expression \(\frac{1}{\log_{3}12} + \frac{1}{\log_{4}12}\), we can use the change of base formula for logarithms and properties of logarithms.

### Step-by-Step Solution:

1. **Apply the Change of Base Formula:**
The change of base formula states that \(\log_{a}b = \frac{\log_{c}b}{\log_{c}a}\). We can use this to rewrite the logarithms in our expression:
\[ \log_{3}12 = \frac{\log 12}{\log 3} \quad \text{and} \quad \log_{4}12 = \frac{\log 12}{\log 4} \]

2. **Rewrite the Expression:**
Substitute the expressions from the change of base formula into our original expression:
\[ \frac{1}{\log_{3}12} + \frac{1}{\log_{4}12} = \frac{\log 3}{\log 12} + \frac{\log 4}{\log 12} \]

3. **Combine the Fractions:**
Since both terms have a common denominator, we can combine them:
\[ \frac{\log 3 + \log 4}{\log 12} \]

4. **Use the Property of Logarithms:**
Recall that \(\log a + \log b = \log(ab)\). Therefore:
\[ \log 3 + \log 4 = \log(3 \cdot 4) = \log 12 \]

5. **Substitute Back:**
Now substitute this back into our expression:
\[ \frac{\log 12}{\log 12} = 1 \]

### Final Answer:
Thus, the value of \(\frac{1}{\log_{3}12} + \frac{1}{\log_{4}12}\) is \(1\).

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