To solve this logarithmic equation, we can use the properties of logarithms to simplify the expression first.

Apply the power rule of logarithms to simplify log₅(x-1) and log₅(x-1)³:

log₅(x-1) - 3log₅(x-1) = -2

Combine the logarithms using the properties of logarithms:

log₅((x-1)/(x-1)³) = -2
log₅(1/(x-1)²) = -2

Rewrite the equation in exponential form:

5^-2 = 1/(x-1)²
1/25 = 1/(x-1)²

Solve for x:

(x-1)² = 25
x - 1 = ±5
x = 1 + 5 or x = 1 - 5
x = 6 or x = -4

Therefore, the solutions to the equation log₅(x-1) - log₅(x-1)³ = -2 are x = 6 and x = -4.