The rate of interest  and principle amount is .

The compound interest can be calculated by the equation .

Where  is number of times interest compounded in an year.

(a)

For annually compounded interest :

Apply ln on both sides.

Take natural logarithm on both sides.

(b)

For monthly compounded interest :

Take natural logarithm on both sides.

(c)

For daily compounded interest :

Take natural logarithm on both sides.

(d)

For continously compounded interest :

Taking this limit as  approaches yeilds the continous compound interest equation .

Take natural logarithm on both sides.

(a) For annually compounded interest time is .

(b) For monthly compounded interest time is .

(c) For daily compounded interest time is .

(d) For continously compounded interest time is .