I've been stuck on interest rates all day?

Question

And it's quite annoying.

A commonly asked question is, How long will it take to double my money? At 10% interest rate and continuous compounding, what is the answer?

My problem is that I don't know how to solve for t because it's in the exponent. So, knowing how to do that would probably help me a lot.

Sorry for all the questions =/

Answer 1

If s is the initial investment, and a is the final amount, r is the fractional linterest rate, and t the time, then:
a = se^(rt)
a / s = e^(rt)
Take logs:
log(a / s) = rt log(e)
t = log(a / s) / r log(e).

The base of the logs here is immaterial, as long as they are both to the same base. If, however, you have logs to base e on your calculator, then since log(e) = 1, the result simplifies to:
t = [ log(base e)(a / s) ] / r.

To double your money at 10%, the time is:
log(base e)(2) / 0.1
= 6.9315 yr.

Answer 2

That is exactly what logarithms are used for, solving for unknown exponents. So since your interest rate is compounded continuously, the formula is A = Pe^rt. The natural log will be used to solve for "t" in the equation because the exponent has base e and the natural log (a.k.a. "ln") also has base e. But I will let you in on a little secret.

Doubling time = ln 2 / interest rate in a decimal (when compounded continuously)

Tripling time = ln 3 / interest rate

Answer 3

with continous compounding and at 10% interest you will double your money in approx 7.2 years, using the rule of 72.

using the exonential formula,

A = Pe^kt, where k is the rate and t is for how long, P is original amount.

if you want to double your money, then A = 2P at that point, so we get:

2P = Pe^kt, which becomes
2 = e^kt, lets puti n the rate of .10

2 = e^.10t. we want to solve for t. take the ln of both sides.

ln 2 = ln e^.1t
ln 2 = .1t
t = ln 2 / .1
t = 10 ln 2
t = 6.9314718 yrs

Answer 4

Yes it is annoying -- you probably have to remember the equation for continuous interest which is:
S=P e^rt (remember pert)
S is the future value, e is base of natural logarithms (denoted as ln rather than log) r is the annual interest rate and t of course is time.

To find time you just take the ln of both sides.
ln S = ln P * rt
so t = (ln S - ln P) / r ( if you want to take only one ln, (ln S - ln P) = ln (S/P).)

so the time it takes to double uses S = 2P so
t = ln (2P/P) / r or t = ln(2)/r [or about 0.7 / r for an estimate.] at 10% that would be 7 years. (calculator ans is 6.93 yr.)

Answer 5

2P=Pe^Yr
2P/P =e^.1Y
2 = e^.1Y
ln2 = .1Y
Y= ln2/.1= 6.93 years

Answer 6

investment = x
x*1.1^t=2x
1.1^t=2
t=log2/log1.1
t=7.273(3.dp)