The value of `(1/(log_(5)210) + 1/(log_(6) 210) + 1/(log_(7)210))` is:

Options

A. 0
B. 1
C. 18
D. 21

Answer

To solve the expression \( \frac{1}{\log_{5} 210} + \frac{1}{\log_{6} 210} + \frac{1}{\log_{7} 210} \), we will 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:
\[ \log_{b} a = \frac{\log_{k} a}{\log_{k} b} \]
where \( k \) is any positive number. We can use this to rewrite each term:
\[ \frac{1}{\log_{5} 210} = \frac{\log_{10} 5}{\log_{10} 210}, \quad \frac{1}{\log_{6} 210} = \frac{\log_{10} 6}{\log_{10} 210}, \quad \frac{1}{\log_{7} 210} = \frac{\log_{10} 7}{\log_{10} 210} \]

2. **Combine the Terms**:
Now, we can combine these fractions:
\[ \frac{1}{\log_{5} 210} + \frac{1}{\log_{6} 210} + \frac{1}{\log_{7} 210} = \frac{\log_{10} 5 + \log_{10} 6 + \log_{10} 7}{\log_{10} 210} \]

3. **Use the Property of Logarithms**:
We know that:
\[ \log_{10} a + \log_{10} b + \log_{10} c = \log_{10} (a \cdot b \cdot c) \]
Therefore:
\[ \log_{10} 5 + \log_{10} 6 + \log_{10} 7 = \log_{10} (5 \cdot 6 \cdot 7) \]

4. **Calculate the Product**:
Now, calculate \( 5 \cdot 6 \cdot 7 \):
\[ 5 \cdot 6 = 30, \quad 30 \cdot 7 = 210 \]
Thus:
\[ \log_{10} (5 \cdot 6 \cdot 7) = \log_{10} 210 \]

5. **Substitute Back**:
Substitute this back into our expression:
\[ \frac{\log_{10} 210}{\log_{10} 210} = 1 \]

### Final Answer:
The value of \( \frac{1}{\log_{5} 210} + \frac{1}{\log_{6} 210} + \frac{1}{\log_{7} 210} \) is \( 1 \).

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