Answer
a) $f^{-1}(n)=3\log_2 \frac{n}{100}$, the time elapsed when there are $n$ bacteria.
b) The population reaches 50000 after approximately 26 hours 54 mins
Work Step by Step
a) $n=f(t)=100\cdot 2^\frac{t}{3}$
$2^\frac{t}{3}=\frac{n}{100}$
$\frac{t}{3}=\log_2 \frac{n}{100}$
$f^{-1}(n)=t=3\log_2 \frac{n}{100}$
This expresses the time elapsed when there are $n$ bacteria.
b) $n=50000,$
$t=3\log_2 \frac{50000}{100}$
$=3\frac{\ln500}{\ln2}$
$\approx26.897$ hours
$\therefore$ the population reaches 50000 after approximately 26 hours 54 mins
