Calculus problem #2: Exponential and Logarithmic Functions. Please Help!!!?

Question

How many years of daily compounding are required to TRIPLE your money if it earns 10% interest?

Answer 1

Brian's solution is incorrect, because he is assuming annual compunding whereas the problem clearly states daily compounding.

Daily compounding: every day your money is multiplied by (1 + 0.10/365). So after n days your money is multiplied by (1 + 0.10/365)^n. We want this to equal 3.
So n = log 3 / log (1 + 0.10/365) = 4010 (nearest integer). So the number of years is 4010/365 = 11.0 years.

Answer 2

y = k (1 + i) ^ x

This is the general formula. Where k is the initial amount, i the percentage of increase or decrease and x the time.

For this problem, since the result is triple the original amount and the percentage of increase is 10% we can write:

3k = k (1 + 0.10) ^ x
3 = (1.1) ^ x

Now we enter logarithms.
To find the solution for x, we want to log both sides with base 1.1:

log 1.1 ^ ( 3 ) = log 1.1 ^ (1.1 ^ x)
log 10 ^ 3 / log 10 ^ 1.1 = x

We change it to base 10 on the left side since most calculators (if not all) utilize logarithms of base 10. So, you would type into your calculator:

[ log ] [ 3 ] / [ log ] [ 1.1 ]

This should result in, approximately 11.53 years.

Answer 3

future value = present value(1+rate)^Time