3 variable simultaneous equation

this problem is really hard. and its hard because i dont know how to write an augmented matrix for it. please explain what you are doing throughout the whole question. of course you can skip the arithmetics. the base of all logs is 10 in this question
determine all real solutions to the system of equations:
x + logx = y - 1
y + log(y-1) = z - 1
z + log(z-2) = x + 2

also prove that there are no more solutions.
thanks in advance

Simplest solution:
make all the logs = 0;

x = 1, y = 2, z = 3

Check:
1 = 2 - 1
2 = 3 - 1
3 = 1 + 2

As for proving that there aren't any more solutions, you could probably do it through substitution, but that seems like it'd take a lot of algebra...

x = z + log(z - 2) - 2
z = y + log(y - 1) - 1

x + log(x) = y - 1

[z + log(z - 2) - 2] + log[z + log(z - 2) - 2] = y - 1

[[y + log(y - 1) - 1] + log([y + log(y - 1) - 1] - 2) - 2] + log[[y + log(y - 1) - 1] + log([y + log(y - 1) - 1] - 2) - 2] = [y + log(y - 1) - 1] - 1

Too lazy to attempt to solve that myself, wolfram says: y ≈ 5.30318049536
Which would make: z ≈ 5.7625348947
And x ≈ 5.0876277991

But plugging this in Wolfram seems like it's just an extraneous solution.

To take a log of a log, you just perform the inside log first, then the outside log

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