Every quarter, your amount increases following the rule
[tex]x\mapsto 1.06x[/tex]
So, after n increments, your amount will be
[tex]x\mapsto (1.06)^nx[/tex]
We want to find the number of increments that will cause our amount to double, i.e. we want to solve for n the following:
[tex](1.06)^nx=2x[/tex]
Simplifying x on both sides, we have
[tex](1.06)^n=2[/tex]
Consider the logarithm base 1.06 of both sides:
[tex]\log_{1.06}((1.06)^n)=\log_{1.06}(2)\iff n\log_{1.06}(1.06)=\log_{1.06}(2) \iff n=\log_{1.06}(2)[/tex]
which we can compute as
[tex]n=\dfrac{\log(2)}{\log(1.06)\approx 12[/tex]
So, after 12 quarters, i.e. 3 years, your amount will be doubled.
