I haven't done one of these in years, so I had to solved it from scratch.
Assuming that dX/dt=kX, we have the separable DE...
integrating we have .
Solving for X(t) we have
From what you said, I believe that
so . Next we solve for k.
But the trick is not to do it completely.
, so .
There is no need to try to approximate k.
The base is here 2, not e. Instead, insert this into our function...
Now check to see if this makes sense. Look at and .
Finally, set , which yields
or , thus .