If `x^(a)=y,y^(b)=z,z^(c)=x` then, abc =

Options

A. xyz
B. `2`
C. `1`
D. `0`

Answer

To solve the problem given by the equations \( x^a = y \), \( y^b = z \), and \( z^c = x \), we will take logarithms and manipulate the equations step by step.

### Step 1: Take logarithms of all equations
Starting with the first equation:
\[ x^a = y \]
Taking logarithms on both sides:
\[ \log(x^a) = \log(y) \]
Using the property of logarithms \( \log(m^n) = n \log(m) \), we get:
\[ a \log(x) = \log(y) \quad \text{(Equation 1)} \]

### Step 2: Take logarithms of the second equation
Now for the second equation:
\[ y^b = z \]
Taking logarithms:
\[ \log(y^b) = \log(z) \]
This gives us:
\[ b \log(y) = \log(z) \quad \text{(Equation 2)} \]

### Step 3: Take logarithms of the third equation
For the third equation:
\[ z^c = x \]
Taking logarithms:
\[ \log(z^c) = \log(x) \]
This results in:
\[ c \log(z) = \log(x) \quad \text{(Equation 3)} \]

### Step 4: Multiply all three equations
Now we will multiply the three equations together:
\[ (a \log(x)) \cdot (b \log(y)) \cdot (c \log(z)) = \log(y) \cdot \log(z) \cdot \log(x) \]
This simplifies to:
\[ abc \cdot \log(x) \cdot \log(y) \cdot \log(z) = \log(x) \cdot \log(y) \cdot \log(z) \]

### Step 5: Cancel out the common terms
Assuming \( \log(x) \cdot \log(y) \cdot \log(z) \neq 0 \), we can divide both sides by \( \log(x) \cdot \log(y) \cdot \log(z) \):
\[ abc = 1 \]

### Final Answer
Thus, we find that:
\[ abc = 1 \]

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