suppose $8000 is deposited in an account with an annual interest rate of 3.5% compounded continuously. find the time required for the value

The formula for continuous compounding:  A  =  P·e^(r·t)

For this problem, let's assume that you start with $1.00 and you end with 20% more, that is, $1.20.

A  =  final amount = 1.20                 P  =  beginning amount = 1.00

r  =  rate (as a decimal) = 0.035        t  =  number of years

--->     1.20  =  1.00·e^(0.035·t)

Divide both sides by 1.00:

--->     1.20  =  e^(0.035·t)

Take the  ln  of both sides:           ln(e^x)  =  x     --->     ln( e^(0.035·t) )  =  0.035t

--->     ln(1.20)  =  0.035·t

Divide both sides by 0.035:

--->  t  =  ln(1.20) / 0.035  =  5.21 years

You don't have to start with $1.00; you can start with any amount, just so you end with a number that is 20% larger.

(This formula is also known as the 'shampoo formula'.)