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...

or

integrating we have .

Solving for X(t) we have

where .

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 .