Logarithmic and Exponential Functions

An exponential function is any mathematical function that can be expressed in the following form: 

f(x)=abx

In this format, a is a constant and is typically the initial value of the function when x = 0.  Recall that anything raised to the power of zero equals 1, so f(0)=ab0 = a(1) = a.  The value b is the base, and it is always greater than zero (b > 0).  The variable is x and it represents the exponent of the function. 

An exponential function can be used to model growth for a wide range of phenomena.  In an exponential growth function, the base (1+r) is always greater than 1 and a > 0.  Exponential growth functions follow this format where r represents the rate of growth: 

f(x)=a(1+r)x

An exponential function can also be used to model decay for a wide range of phenomena.   Exponential decay functions follow this format: 

f(x)=a(1-r)x

In this exponential decay function, the base (1-r) is always less than 1, a > 0, and r represents the rate of decay.  Another option is for the exponent to be negative.  In that case, the exponential decay function will follow this format: 

f(x)=ab-x

In this version of the exponential decay function, b > 1 and a > 0. 

Logarithmic functions have the following format:  f(x)=logbx

The following videos cover the properties of exponential and logarithmic functions and the types of problems they are typically involved in solving.  A complete transcript and notes for these videos is provided here: 

Modeling growth and decay is a common activity when dealing with data collected from the natural world.  In the following video, you will learn how exponential functions are used to model growth and decay.  In general, exponential functions have the form 𝑓(𝑥) = 𝑎^𝑥 where 𝑎 is a population constant referred to as the base and 𝑥 is the variable.  The following video demonstrates the basic features of exponential functions. 

The video below continues the exploration into the nature of exponential functions.  A common base that occurs in many exponential models is Euler’s number 𝑒.  Because it can not be represented by a fraction, 𝑒 is an irrational number that is approximately equal to 2.71828.  

In the following video, the characteristics and applications of logarithmic functions are demonstrated.