Combine, Then Undo

log_6(x) + log_6(x - 1) = 1 has two logs, and the undo move works on one. So join them first with the product rule, which holds while both inputs are positive: log_6(x) + log_6(x - 1) = log_6(x(x - 1)).

Now there is one log, and raising 6 to both sides undoes it: x(x - 1) = 6, an ordinary quadratic. Expand, move the 6 across and factor: (x - 3)(x + 2) = 0, so x is 3 or -2.

Those are only candidates. Put each back into the original equation. With x = 3 the inputs are 3 and 2, both positive, and log_6(3) + log_6(2) is log_6(6), which is 1. With x = -2 the first input is -2, and the log of a negative number has no value, so -2 is thrown out. The board's move Check x = ... in the original equation runs this test for each candidate.