Module 4 – Exponential and Logarithmic Functions - Introductory Examples

Population Growth

 

“Exponential Growth” is a phrase that we all see almost daily in the news, but what does it mean and why is it so common?

 

In any population of living things, each year some die and others are born. If there is plenty of space and food available, then the number of each is roughly proportional to the size of the population, so the population gets multiplied by the same growth factor each year (more detail). So at a time t years after we start watching, it is given by  (where  is the population we started with and the constant a is the annual growth factor).

 

Here, the formula for  has the variable t as the exponent (‘upstairs’ part) in a power expression.  Up to now, we have only considered power expressions in which the exponents are constants (e.g. polynomials like  ), but in this unit we study functions defined by equations involving variables in the exponents. Such functions are called exponential functions (or logarithmic functions if it is the dependent variable in the exponent).