Solution:

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

To find: xyz

Let's solve the given equations step by step.

1. Finding the value of a in terms of c

c^z = a

Taking logarithm on both sides,

zlog(c) = log(a)

a = c^(log(a)/log(c))

2. Finding the value of b in terms of c

b^y = c

Taking logarithm on both sides,

ylog(b) = log(c)

b = c^(log(c)/log(b))

3. Substituting the value of a and b in the first equation

a^x = b

Putting the values of a and b

(c^(log(a)/log(c)))^x = (c^(log(c)/log(b)))

c^(x*log(a)/log(c)) = c^(log(c)/log(b))

Equating the powers of c on both sides,

x*log(a)/log(c) = log(c)/log(b)

log(a)/log(c) = log(c)/log(b*y)

log(a) * log(b*y) = log(c)^2

4. Finding the value of xyz

Multiplying the given equations,

a^x * b^y * c^z = b*c*a

Substituting the values of a and b from the above equations, we get

c^(x*log(a)/log(c)) * c^(y*log(b)/log(c)) * c^z = c*c^(log(a)/log(c)) * c^(log(c)/log(b))

c^(x*log(a)/log(c)+y*log(b)/log(c)+z) = c^(1+log(a)/log(b))

Equating the powers of c on both sides,

x*log(a)/log(c) + y*log(b)/log(c) + z = 1 + log(a)/log(b)

Substituting the value of log(a)/log(c) and log(b)/log(c) from the equations (3) and (4) respectively, we get

x*log(a)*log(b*y)/log(c)^2 + y*log(c)*log(a)/log(c)^2 + z = 1 + log(a)*log(c)/log(b)*log(c)

Simplifying the above equation, we get

xyz = 1

Hence, the value of xyz is 1.