If `x ="log"_(a)(bc), y ="log"_(b)(ca) " and "z = "log"_(c)(ab), ` then which of the following is correct?

Options

A. `x + y +z =1`
B. `(1)/(1+x) + (1)/(1+y) + (1)/(1+z) =1`
C. xyz =1
D. none of these

Answer

To solve the problem, we need to analyze the expressions given for \( x \), \( y \), and \( z \) and determine which of the provided options is correct.

Given:
- \( x = \log_a(bc) \)
- \( y = \log_b(ca) \)
- \( z = \log_c(ab) \)

### Step 1: Evaluate \( x + y + z \)

We start with the first option, which states that \( x + y + z = 1 \).

\[ x + y + z = \log_a(bc) + \log_b(ca) + \log_c(ab) \]

Using the change of base formula, we can express each logarithm in terms of natural logarithms (or any common base):

\[ x = \frac{\log(bc)}{\log(a)}, \quad y = \frac{\log(ca)}{\log(b)}, \quad z = \frac{\log(ab)}{\log(c)} \]

Thus,

\[ x + y + z = \frac{\log(bc)}{\log(a)} + \frac{\log(ca)}{\log(b)} + \frac{\log(ab)}{\log(c)} \]

This expression cannot be simplified to show that it equals 1. Therefore, we conclude that the first option is **not correct**.

### Step 2: Evaluate \( \frac{1}{1+x} + \frac{1}{1+y} + \frac{1}{1+z} \)

Next, we check the second option:

\[ \frac{1}{1+x} + \frac{1}{1+y} + \frac{1}{1+z} \]

Calculating \( \frac{1}{1+x} \):

\[ \frac{1}{1+x} = \frac{1}{1+\log_a(bc)} = \frac{1}{\frac{\log(a)}{\log(a)} + \frac{\log(bc)}{\log(a)}} = \frac{\log(a)}{\log(a) + \log(bc)} = \frac{\log(a)}{\log(a) + \log(b) + \log(c)} \]

Similarly, we can find:

\[ \frac{1}{1+y} = \frac{\log(b)}{\log(b) + \log(c) + \log(a)} \]
\[ \frac{1}{1+z} = \frac{\log(c)}{\log(c) + \log(a) + \log(b)} \]

Adding these together:

\[ \frac{\log(a)}{\log(a) + \log(b) + \log(c)} + \frac{\log(b)}{\log(a) + \log(b) + \log(c)} + \frac{\log(c)}{\log(a) + \log(b) + \log(c)} = \frac{\log(a) + \log(b) + \log(c)}{\log(a) + \log(b) + \log(c)} = 1 \]

Thus, the second option is **correct**.

### Step 3: Evaluate \( xyz \)

Now, we check the third option which states \( xyz = 1 \):

\[ xyz = \log_a(bc) \cdot \log_b(ca) \cdot \log_c(ab) \]

Using the change of base formula again, we can express this as:

\[ xyz = \left(\frac{\log(bc)}{\log(a)}\right) \cdot \left(\frac{\log(ca)}{\log(b)}\right) \cdot \left(\frac{\log(ab)}{\log(c)}\right) \]

This expression does not simplify to 1. Therefore, the third option is **not correct**.

### Conclusion

After evaluating all options, we find that the only correct option is:

**Option 2: \( \frac{1}{1+x} + \frac{1}{1+y} + \frac{1}{1+z} = 1 \)**