`x^((log_x)log_a ylog_y z)` is equal to

Options

A. x
B. y
C. z
D. a

Answer

To solve the expression \( x^{(\log_x (\log_a y) \log_y z)} \), we will simplify it step by step.

### Step 1: Rewrite the logarithmic expressions
We start with the expression:
\[ x^{(\log_x (\log_a y) \log_y z)} \]
Using the change of base formula for logarithms, we can rewrite \( \log_y z \) as:
\[ \log_y z = \frac{\log_a z}{\log_a y} \]
Thus, we can substitute this into the original expression:
\[ x^{(\log_x (\log_a y) \cdot \frac{\log_a z}{\log_a y})} \]

### Step 2: Simplify the expression
Now, we can simplify \( \log_x (\log_a y) \cdot \frac{\log_a z}{\log_a y} \):
\[ \log_x (\log_a y) \cdot \frac{\log_a z}{\log_a y} = \frac{\log_a (\log_a y)}{\log_a x} \cdot \frac{\log_a z}{\log_a y} \]
This can be rewritten as:
\[ \frac{\log_a (\log_a y) \cdot \log_a z}{\log_a x \cdot \log_a y} \]

### Step 3: Substitute back into the expression
Now we substitute this back into our expression:
\[ x^{\left(\frac{\log_a (\log_a y) \cdot \log_a z}{\log_a x \cdot \log_a y}\right)} \]

### Step 4: Apply properties of logarithms
Using the property \( a^{\log_b c} = c^{\log_b a} \), we can rewrite the expression:
\[ x^{\left(\frac{\log_a z}{\log_a y}\right)} = z^{\left(\frac{\log_a x}{\log_a y}\right)} \]

### Step 5: Final simplification
Now, we can simplify further:
\[ z^{\left(\frac{\log_a x}{\log_a y}\right)} = z^{\log_y x} \]
Using the property of logarithms, we can express this as:
\[ z^{\log_y x} = x^{\log_y z} \]

### Conclusion
Thus, the final simplified expression is:
\[ \boxed{z^{\log_y x}} \]

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