Well, we just have to take the simple interest formula, which says what amount P do you have when you have a rate i, an initial amount A, and a time in years n:

P = A(1 + ni)

Which is close, but not exactly what you got. I could not find a single equation that had (1 + i)^n.

So we plug the values above into the equation above:

$20,000 = $10,000 (1 + .15n)

$20,000 = $10,000 + $1500n

$10,000 = $1500n

6.667 years = n

So that's how long it will take $10,000 to grow to $20,000.