how long does it take for an investment to double in value if it is invested at 10% compounded monthly question mark monthly? compounded continuously?

The first step to determining the answer to this item is to calculate for the effective interest using the equation,

                  ieff = (1 + i/m)^m - 1
where ieff is the effective interest, i is the given interest and m is the number of compounding period.

Part A: m in this item is equal to 12.
Substituting,
                ieff = (1 + 0.10/12)^12 - 1 = 0.1047

The amount of money after n years is calculated through the equation,
               An = A(1 + ieff)^n
If An/A = 2 then,
                  2 = (1 + 0.1047)^n
The value of n is 6.96 years

Part B: For the continuously compounding,
               An = Ae^(rt)
           An/A = 2 = e^(0.10t)
The value of t is equal to 6.93 years. 

Hence, the answers:
Part A: 6.96 years
Part B: 6.93 years