Logarithm functions

Question

1.Find the inverse of A(t) = 500(1/2)^t/272

2.What does the input variable of the inverse function represent?

3.What does the output variable of the new function represent?

4.If 100 grams of cobalt-57 has decayed to 62 grams, how much time has passed?

Answer 1

y is the amount decay and t is the time;

the inverse gives the time required for the specified amount of decay

y = A(t) = 500(1/2)^(t/272)

t = 500 * (1/2)^(y/272) <--- swaps domain and range

(t/500) = (1/2)^(y/272) <--- must solve for y

log( t/500) = (y/272) * log (1/2) <--- log of both sides; exponent moves in front

log(t/500) / log(1/2) = y/272 <--- isolates the y/272 by dividing both sides by log(1/2)

(log t - log 500)/( log 1 - log 2) = y / 272 <--- property of logs : log (a/b) = log a - log b

272*(log t - log 500)/ -log 2 = y <--- multiplies both sides by 272; also log 1 = 0

so A_inverse( x) = 272*(log x - log 500) / -log 2 where x is the amount of decay and the

out is the time it takes for such decay

x=62

272 * (log 62 - log 500)/ -log2 = 819.151929+ units of time