Horton equation:

f = f_c + (f_o - f_c) e^-kt

f - f_c = (f_o - f_c) e^-kt

2 - 0.5 = (f_o - 0.5) e^-k

1 - 0.5 = (f_o - 0.5) e^-2k

1.5 = (f_o - 0.5) e^-k

0.5 = (f_o - 0.5) e^-2k

Dividing these two equations:

3 = e^k

k = ln (3) = 1.0986 hr^-1    ANSWER.

Solving for f_o:

f_o = 1.5 e^k + 0.5

f_o = (1.5 × 3) + 0.5 = 5 mm/hr    ANSWER.

f_c = 0.5 mm/hr    ANSWER.

Verification:

f = 0.5 + (5.0 - 0.5) e^-1.0986 = 2 mm/hr   OK

f = 0.5 + (5.0 - 0.5) e^{-2 × 1.0986} = 1 mm/hr   OK