By inspection 6 = 2*3^1 and 54 = 2*3^3, so, (one version of) the formula is

f = 2*3^t

f(t) = Aoe^(kt)

6 = Ao*e^k

54 = Aoe^(3k)

9 = e^(2k)

2k = ln 9

k = 1.0986

f(t) = Ao*e^(1.096t)

6 = Ao*e^1.096

Ao = 2.0052

f(t) = 2.0052e^(1.096t)

You probably mean this form or something equivalent:

f(t) = ab^t, where b > 0

f(1) = 6

ab¹ = 6

ab = 6

f(3) = 54

ab³ = 54

(ab³)/(ab) = 54/6

b² = 9

b = 3

ab = 6

a(3) = 6

a = 2

f(t) = 2(3)^t