Population growth logarithm

The question is: The bacterial population in a certain culture is given by where is in hours and is the initial population (when ). Determine how long it takes for this population to double in size.

So, from this, i've worked out that and then i took a stab at rearranging the equation into the form and got

Now, although I *believe* this to be the right answer something isn't ringing true since natural logarithms didn't come into it at all. I've always been told that population growth was one of the main things natural logs were for outside of calculus.

Any help or suggestions would be greatly appreciated.