Given:
a^x = b
b^y = c
c^z = a

To find:
The value of xyz

Solution:

Step 1: Solving the equations
We can start by solving the given equations one by one.

Solving a^x = b:
Taking the logarithm of both sides with base a:
loga(a^x) = loga(b)
x*loga(a) = loga(b)
x*1 = loga(b)
x = loga(b)

Solving b^y = c:
Taking the logarithm of both sides with base b:
logb(b^y) = logb(c)
y*logb(b) = logb(c)
y*1 = logb(c)
y = logb(c)

Solving c^z = a:
Taking the logarithm of both sides with base c:
logc(c^z) = logc(a)
z*logc(c) = logc(a)
z*1 = logc(a)
z = logc(a)

Step 2: Finding the value of xyz
Now, substituting the values of x, y, and z obtained from the above solutions, we get:
xyz = (loga(b)) * (logb(c)) * (logc(a))

Step 3: Simplifying the expression
Using the property of logarithms, we can simplify the expression as follows:
xyz = (loga(b)) * (logb(c)) * (logc(a))
= loga(b) * logb(c) * logc(a)
= loga(c) * logc(a) * logb(c)
= loga(c) * logb(c) * logc(a)
= loga(c) * logb(c) * (1/z)
= loga(c) * (1/y) * (1/z)
= 1/x * 1/y * 1/z
= 1/(xyz)

Step 4: Solving for xyz
Now, we have the equation:
xyz = 1/(xyz)

Multiplying both sides by (xyz), we get:
(xyz)^2 = 1

Taking the square root of both sides, we get:
xyz = ±1

Therefore, the value of xyz can be either 1 or -1.

Conclusion:
The value of xyz is either 1 or -1, depending on the given equations.