Lesson 7: Modeling with Exponential and Logarithmic Functions

Property

A mathematical model is a simplified description of reality that uses mathematics to help us understand a system or process. We can represent the relationship between variables by:

• a table of values
• a graph
• an algebraic equation

Examples

For a bike rental with a 5 dollar fee and 3 dollars per hour rate, the cost $C$ for time $t$ can be shown as a table: (0, 5), (1, 8), (2, 11).

Property

An exponential function has the form

The constant $a$ is the $y$-intercept of the graph because $f(0) = a \cdot b^0 = a \cdot 1 = a$.
The positive constant $b$ is called the base. We do not allow $b$ to be negative, because if $b < 0$, then $b^x$ is not a real number for some values of $x$. We also exclude $b = 1$ because $1^x = 1$ for all values of $x$, which is a constant function.

Examples

• The function $f(x) = 5(2)^x$ is an exponential function where the initial value is $a=5$ and the growth factor is the base $b=2$.

• The function $P(t) = 100(0.75)^t$ represents exponential decay with an initial amount of $100$ and a decay factor of $0.75$.