The value of `[(1)/(log_(xy)(xyz))+(1)/(log_(yz)(xyz))+(1)/(log_(zx)(xyz))]` is equal to

Options

A. A)1
B. B)2
C. C)3
D. D)4

Answer

To solve the expression \(\left[\frac{1}{\log_{xy}(xyz)} + \frac{1}{\log_{yz}(xyz)} + \frac{1}{\log_{zx}(xyz)}\right]\), we will use the change of base formula for logarithms, which states that:

\[ \log_a(b) = \frac{\log_c(b)}{\log_c(a)} \]

for any base \(c\).

### Step 1: Apply the change of base formula
We can rewrite each term in the expression using the change of base formula. Let's choose base \(xyz\):

\[ \log_{xy}(xyz) = \frac{\log_{xyz}(xyz)}{\log_{xyz}(xy)} = \frac{1}{\log_{xyz}(xy)} \]

\[ \log_{yz}(xyz) = \frac{\log_{xyz}(xyz)}{\log_{xyz}(yz)} = \frac{1}{\log_{xyz}(yz)} \]

\[ \log_{zx}(xyz) = \frac{\log_{xyz}(xyz)}{\log_{xyz}(zx)} = \frac{1}{\log_{xyz}(zx)} \]

### Step 2: Substitute back into the expression
Now we can substitute these back into the original expression:

\[ \frac{1}{\log_{xy}(xyz)} = \log_{xyz}(xy) \]
\[ \frac{1}{\log_{yz}(xyz)} = \log_{xyz}(yz) \]
\[ \frac{1}{\log_{zx}(xyz)} = \log_{xyz}(zx) \]

Thus, the expression becomes:

\[ \log_{xyz}(xy) + \log_{xyz}(yz) + \log_{xyz}(zx) \]

### Step 3: Combine the logarithms
Using the property of logarithms that states \(\log_a(b) + \log_a(c) = \log_a(bc)\), we can combine these:

\[ \log_{xyz}(xy) + \log_{xyz}(yz) + \log_{xyz}(zx) = \log_{xyz}(xy \cdot yz \cdot zx) \]

### Step 4: Simplify the argument of the logarithm
Now we simplify the argument \(xy \cdot yz \cdot zx\):

\[ xy \cdot yz \cdot zx = x^1y^2z^1 \]

Thus, we have:

\[ \log_{xyz}(xyz \cdot xyz) = \log_{xyz}((xyz)^2) \]

### Step 5: Final simplification
By the property of logarithms, we can simplify this further:

\[ \log_{xyz}((xyz)^2) = 2 \cdot \log_{xyz}(xyz) = 2 \]

### Final Result
Therefore, the value of the original expression is:

\[ \boxed{2} \]