the formula for growth is:
f = p * (1 + r)^n
r is the growth rate per time period.
n is the number of time periods.
p is the present value
f is the future value.
in your problem:
f = 32
p = 256
r = -.5
1+r = .5
n is what you have to solve for.
the equation becomes:
32 = 298 * (.5)^n
divide both sides of this equation by 298 to get:
32/298 = .5^n
take the log of both sides of this equation to get:
log(32/298) = log(.5^n)
since log(.5^n) = n * log(.5), your equation becomes:
log(32/298) = n * log(.5)
divide both sides of this equation by log(.5) to get:
log(32/298) / log(.5) = n
use your calculator to solve for n to get:
n = 3.21916852...
store that in your calculator and use it to confirm the solution is correct.
your original equation becomes:
32 = 298 * (.5)^3.21916852...
you get 32 = 32
this confirms your calculations are correct.
256 grams of the substance will be reduced to 32 grams in 3.21916852... days if it loses half its weight each day.
start with 298.
at end of 1 day, you have 149
at end of 2 days, you have 74.5
at end of 3 days, you have 37.25
at end of 3.21916852... days, you have 32.
that fractional part of the last day is calculated as follows:
37.25 * .5^.21916852... = 32
after 10 days, you will be left with 298 * .5^10 = .291015625 grams.
you can see the day by day effect by looking at the following table:
day grams
0 298
1 149
2 74.5
3 37.25
3.2191... 32
4 18.625
5 9.3125
6 4.6563
7 2.23281
8 1.1641
9 .58203
10 .29102