Answer:
t = log(2)/0.12 years (exact value)
= 5.776 years (to three decimal places)
Explanation:
The future value of an amount P at continuous interest rate i% for t years can be calculated using the continuous interest formula:
FV = Pe^((i)t)
where e is the Euler's constant = 2.7182818284....
We're given
10000 = 5000 e^((0.12)t)
simplify
2 = e^(0.12t) ........................ see below
[ please compare if this is the same as the one shown in the original question, the formula posted as "2 = e 120" does not look right, probably because of typographical difficulties, then answer if the equation given in the question can be used to solve the problem]
take (natural) log using the power property of logarithms
log(2) = 0.12t log(e)
using natural log, log(e) = 1
log(2) = 0.12t
simplify
t = log(2)/0.12 years (exact value)
To obtain the time (in years)
t = log(2)/0.12 = 0.6931/0.12 = 5.7762 years
Check using the rule of 72 (approximate)
72/12(%) = 6,
Since continuous interest accumulates interest faster, so 5.8 years sound reasonable.