If `a^(x)=b,b^(y)=c,c^(z)=a`, then value of xyz is

Options

A. `0`
B. `1`
C. `2`
D. `3`

Answer

To solve the problem, we start with the given equations:

1. \( a^x = b \)
2. \( b^y = c \)
3. \( c^z = a \)

We want to find the value of \( xyz \).

### Step 1: Express \( b \), \( c \), and \( a \) in terms of logarithms

From the first equation \( a^x = b \), we can take logarithm on both sides:

\[ \log_a b = x \]

From the second equation \( b^y = c \):

\[ \log_b c = y \]

From the third equation \( c^z = a \):

\[ \log_c a = z \]

### Step 2: Use the change of base formula

Using the change of base formula for logarithms, we can express \( y \) and \( z \) in terms of base \( a \):

1. For \( y \):
\[ y = \log_b c = \frac{\log_a c}{\log_a b} \]

2. For \( z \):
\[ z = \log_c a = \frac{\log_a a}{\log_a c} = \frac{1}{\log_a c} \]

### Step 3: Substitute back into the expression for \( xyz \)

Now we can express \( xyz \):

\[ xyz = \log_a b \cdot \frac{\log_a c}{\log_a b} \cdot \frac{1}{\log_a c} \]

### Step 4: Simplify the expression

Notice that \( \log_a b \) in the numerator and denominator will cancel out, and \( \log_a c \) in the numerator and denominator will also cancel out:

\[ xyz = 1 \]

### Conclusion

Thus, the value of \( xyz \) is:

\[ \boxed{1} \]

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