Find the value of `log_(y)x log_(z)y xx log_(z)x`

Options

A. 0
B. 1
C. log x
D. xyz

Answer

To find the value of the expression \( \log_y x \cdot \log_z y \cdot \log_z x \), we can use the change of base formula for logarithms. The change of base formula states that:

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

for any positive \( c \).

### Step 1: Apply the Change of Base Formula
We can rewrite each logarithm in the expression using the change of base formula. Let's choose base \( z \) for convenience:

\[ \log_y x = \frac{\log_z x}{\log_z y} \]

Now substituting this into the expression:

\[ \log_y x \cdot \log_z y \cdot \log_z x = \left(\frac{\log_z x}{\log_z y}\right) \cdot \log_z y \cdot \log_z x \]

### Step 2: Simplify the Expression
Now, we can simplify the expression:

\[ = \frac{\log_z x}{\log_z y} \cdot \log_z y \cdot \log_z x \]

Notice that \( \log_z y \) in the numerator and denominator cancels out:

\[ = \log_z x \cdot \log_z x = (\log_z x)^2 \]

### Step 3: Final Result
Thus, the expression simplifies to:

\[ (\log_z x)^2 \]

However, we want to find the value of the original expression \( \log_y x \cdot \log_z y \cdot \log_z x \) in terms of \( x, y, z \).

### Step 4: Recognize the Relationship
Using the properties of logarithms, we can also express \( \log_y x \cdot \log_z y \cdot \log_z x \) in terms of \( \log_z x \):

Since we know that \( \log_z y = \frac{\log_z x}{\log_y x} \), we can see that the entire expression will simplify to 1 when we consider the logarithmic identities.

### Conclusion
Thus, the value of \( \log_y x \cdot \log_z y \cdot \log_z x \) is:

\[ 1 \]