How long will it take $500.00 to grow to $2,000.00 at an interest rate of 4% if
the interest is compounded continuously.

All exponential growth or decay problems come from this formula:

A = Pert

Where P is the original amount, and A is the final amount, and r is the
yearly rate for continuous compounding problems.

So we substitute A = 2000, P = 500, and r - .04 and solve for t:

2000 = 500e.04t

Divide both sides by 500

4 = e.04t

Rewrite that using the rule:

The exponential equation Y = eX can be rewritten as the 
logarithmic equation X = ln(Y).

.04t = ln(4)

Divide both sides by .04

  t = ln(4)/.04 = 1.386294361/.04 = 34.65735903

or more than 34 1/2 years.  By the way, things that

cost $500 34 years ago cost a lot MORE THAN $2000 today.

In 1972, I paid $30,000 for a house that is worth $250,000
today!!!!!!

Moral of the story: Investing at 4% interest does not
come close to keeping up with inflation.

Edwin