Exponential functions are useful for predicting the growth of something after a certain amount of time and it has many real-world applications in science and lives. One such example is the ability to accurately predict bacterial growth, that is the asexual reproduction of a bacterium into two daughter cells repeatedly, over time. To approach this problem, the fundamental principles of exponential functions needs to be explained.

In mathematics, exponential functions differ from polynomial functions such that the former has the independent variable raised as an exponent. Therefore, the simplest exponential function is described as f(x)=ab^x where

*Problem: A biologist is researching a newly-discovered species of bacteria. At time *

*0 hours, he puts one hundred bacteria into what he has determined to be a favorable growth medium. Six hours later, he measures 450 bacteria. Assuming exponential growth, what is the growth constant*

*for the bacteria? (Round k to two decimal places.)*

For these types of problems, it is important to recognize the unit of time (here hours), so that we keep all of our units the same when answering the question. The question tells us that when the time is zero,

*50 = 100e^6k 4.5 = e^6k*

*n(4.5)/ = 6k ln(4.5/6 = k*

*= 0.250679566129*

Therefore, the growth constant k or rate in this problem is