Apply the formula for compound interest
A = P(1+r)x
where P is the invested money, r- monthly rate, x - number of months. According to the problem A = 2P. Thus, if wedivide both sides of the formula by P we will obtain
2 = (1+r)x
Take logarithms of both sides: ln2 = x ln(1+r). Thus
x = ln2/ln(1+r)
r = 0.06/12 = 0.005. ln(1+0.005) = 0.005 (0.00498.. more precisely). Hense, it follows x = ln2/0.005 = 138.6 months = 11.55 years.