Check out this dialogue between two calculus students (based on a true story):

Devyn
Riley, I’ve been thinking about the interest on my bank account.
Riley
So, like the compound interest formula,
Devyn
Yes! To make it easier to work with, suppose I deposit just $1, so .
Riley
A $1 bank account balance? I can imagine that!
Devyn
If the interest is compounded monthly, , and if the deposit is at interest, then .
Riley
Since , that brings you to
Devyn
My question is: How long does it take the account to double in value? How long until the account balance is $2?
Riley
Can’t we just setup
Devyn
Of course, but how to solve that for ? The variable is up in an exponent.
Riley
Hmmmm. I’m not sure…
What kind of operation will allow us to solve for ?
Take the square root of both sides. Take the cosecent of both sides. Raise both sides to the -th power. Take a logarithm of both sides.
Take the compound interest formula above with , , and : Plug in larger and larger values of and see what happens to the values of .
tends to , so . The exponent is getting bigger and bigger, so . . .