When the interest is compounded k times per year, with i / 100 replaced by r and A/P = 2, the formula you have becomes:

2 = (1 + r/k)^(kt)

Expanding the RHS using the binomial theorem gives:

2 = 1 + (kt)(r/k) + (kt)(kt – 1) (r/k)^2 / 2!

+ (kt)(kt – 1)(kt – 2) (r/k) / 3! + …

2 = 1 + tr + t(t – 1/k)(r^2 / 2!) + t(t – 1/k)(t – 2/k)(r^3 / 3!) + …

As k tends to infinity, 1/k tends to 0, and this becomes:

2 = 1 + tr + (tr)^2 / 2! + (tr)^3 / 3! + …

The RHS of this formula is the power series for e^(rt).

Hence:

2 = e^(rt)

rt = log(base e) 2

t = [ log(base e)2 ] / r

t = 0.6913 / r

Substituting 100i back for r:

t = 69.13 / i

but I don’t really see the justification for 70.

However this is the basis of the derivation in the link below, where at the end the article just says ’70 is close enough’