P(t)=Pe^0.05t is correct.

Q:

Since the original investment is $3,000.00, doubling means that the final balance is $6,000.00. To find out how long it takes for this to happen ( i.e. to find t ), plug in P = 3000, P(t) = 6000, and r = 0.05 in the continuous compound interest formula, and solve for t.

Doing this, we get,

3000 * e^(0.05t) = 6000

or e^(0.05t) = 6000/3000 = 2

So we have to solve the exponential equation, e^(0.05t) = 2, by converting it into log (ln) notation. This will give us,

0.05t = ln 2

t = ln2 / 0.05 = 0.693147180559945 / 0.05

t = 13.86294361 years

Hope this helps.

6000=3000e^0.05t

2 = e^0.05t

Ln(2) = 0.05t

0.6931/0.05 = t

t = 13.9 years

3000 * 2 = 3000 * e^0.05t

2 = e^0.05t

ln[2] = ln[e^0.05t]

ln[2] = 0.05t * ln[e]

ln[2] = 0.05t * 1

ln[2]/0.05 = t

13.9 years = t

6000=3000e^.05t

2=e^.05t

ln2=lne^.05t

ln2=.05t

.6931=.05t

13.9=t

in about 13.9 years