exponential & logarithmic functions

Exponential & Logarithmic Functions

Exponential functions and logarithm functions are important in both theory and practice. In this unit we look at the graphs of exponential and logarithm functions, and see how they are related. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. After reading this text, and/or viewing the video tutorial on this topic, you should be able to: 
• specify for which values of a the exponential function f(x) = a x may be defined, 
• recognize the domain and range of an exponential function, 
• identify a particular point which is on the graph of every exponential function, 
• specify for which values of a the logarithm function f(x) = loga x may be defined, 
• recognize the domain and range of a logarithm function, 
• identify a particular point which is on the graph of every logarithm function, 
• understand the relationship between the exponential function f(x) = e x and the natural logarithm function f(x) = ln x. 

Exponential Graphs:

Once you know the shape of an exponential graph , you can shift it vertically or horizontally, stretch it, shrink it, reflect it, check answers with it, and most important interpret the graph.
Example 1:
The function  is always positive. There is simply no value of x that will cause the value of  to be negative. What does this mean in terms of a graph? It means that the entire graph of the function  is located in quadrants I and II.
Graph the function  . Notice that the graph never crosses the x-axis. Why is that so? It is because there is no value of x that will cause the value of f(x) in the formula  to equal 0.
Notice that the graph crosses the y-axis at 1. Why is that so? The value of x is always zero on the y-axis. Substitute 0 for x in the equation  :  . This translates to the point (0, 1).
Notice on the graph that, as the value of x increases, the value of f(x) also increases. This means that the function is an increasing function. Recall that an increasing function is a one-to-one-function, and a one-to-one function has a unique inverse.
The inverse of an exponential function is a logarithmic function and the inverse of a logarithmic function is an exponential function.
Notice also on the graph that as x gets larger and larger, the function value of f(x) is increasing more and more dramatically. This is why the function is called an exponential function.

Logarithmic Graphs:

Once you know the shape of a logarithmic graph , you can shift it vertically or horizontally, stretch it, shrink it, reflect it, check answers with it, and most important interpret the graph.
Example 2:
Graph the function  . Notice that the graph of this function is located entirely in quadrants I and IV. Notice also that the graph never touches the y-axis.
What does that mean? It means that the value of x (domain of the function f(x) in the equation  is always positive. Why is this so? Recall that the equation can be rewritten as the exponential function  . There is no value of f(x) that can cause the value of x to be negative or zero.
The graph of  will never cross the y-axis because x can never equal 0. The graph will always cross the x-axis at 1.
Notice on the graph that, as x increases, the f(x) also increases. This means that the function is an increasing function. Recall that an increasing function is a one-to-one-function, and a one-to-one function has a unique inverse.
Notice on the graph that the increase in the value of the function is most dramatic between 0 and 1. After x = 1, as x gets larger and larger, the increasing function values begin to slow down (the increase get smaller and smaller as x gets larger and larger).
Notice on the graph that the function values are positive for x's that are greater than 1 and negative for x's less than 1.