Given information:
- log_x(z) = 4
- log_10(y z) = 2
- log_(xy)(z) = 4/7

Calculating x, y, and z:
- From log_x(z) = 4, we can rewrite it as x^4 = z
- From log_10(y z) = 2, we can rewrite it as 10^2 = y z
- Simplifying 10^2 = y z to get y z = 100
- Since x^4 = z, we substitute z with x^4 in y z = 100 to get y x^4 = 100
- From log_(xy)(z) = 4/7, we can rewrite it as (xy)^(4/7) = z
- Substitute z with x^4 in (xy)^(4/7) = x^4 to get (xy)^(4/7) = x^4

Solving for x, y, and z:
- From (xy)^(4/7) = x^4, we can rewrite it as y^(4/7) = x^(4 - 4/7)
- Simplifying y^(4/7) = x^(4 - 4/7) to get y^(4/7) = x^(24/7 - 4/7) = x^(20/7)
- Therefore, y = x^(5)
- Substitute y = x^(5) into y x^4 = 100 to get x^(5) x^4 = 100
- Simplifying x^(9) = 100 to get x = 100^(1/9)
- Substitute x = 100^(1/9) into y = x^(5) to get y = (100^(1/9))^5 = 100^(5/9)
- Finally, z = x^4 = (100^(1/9))^4 = 100^(4/9)

Calculating z^-2:
- z^(-2) = (100^(4/9))^(-2) = 100^(-8/9) = 1/100^(8/9) = 1/100^(8/9) = 1/10^(8/3) = 1/10^(2.67) ≈ 0.001995
Therefore, z^-2 is approximately 0.001995.