Logarithmic Functions
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Definition:
The logarithmic function is the function , where b is any number such that .
                  is equivalent to
                The function is read "log base b of x".             


Let's examine the function
 
Check out the table
and the graph. 

 Remember that x must be positive.
x f (x) = y
1/4
1/2
1
2
3
4


Most logarithmic graphs resemble this same shape.
This graph is very, very close to the y-axis but does not cross it.  The graph increases as it progresses to the right (as seen in the graph at the right).

In a straight line, the "rate of change" remains the same across the graph.  In these graphs, the "rate of change" increases or decreases across the graphs.
 

Characteristics:
     Such logarithmic graphs of the form   have certain characteristics in common:

    Logarithmic functions are one-to-one functions.		   •  graph crosses the x-axis at (1, 0)

•  when b > 1, the graph increases

•  when 0 < b < 1, the graph decreases

•  the domain is all positive real numbers (never zero)

•  the range is all real numbers

•  graph passes the vertical line test - it is a function

•  graph passes the horizontal line test - its inverse is also a function.

•  graph is asymptotic to the y-axis - gets very, very close to the y-axis but does not touch it or cross it.
 

 
Natural Logarithmic Function:

The function defined by

is called the natural logarithmic function.

(e is an irrational number, approximately 2.71828183, named after the 18th century Swiss mathematician, Leonhard Euler .)

Notice how the characteristics of this graph are similar to those seen above.

 This function is simply a "version" of
   where  b >1.


While the graph may "appear" to STOP near -4 on the y-axis, it does NOT stop.  It continues extremely close to the y-axis heading to negative infinity.

  

Inverse of  :

Since is a one-to-one function, we know that its inverse will also be a function. 

When we graph the inverse of the natural logarithmic function, we notice that we obtain the natural exponential function, f (x) = ex.

Notice how (1,0) from  y = ln x  becomes (0,1) for  f (x) = ex.  The coordinates switch places between a graph and its inverse.

Other Logarithmic Inverses:

Graph Inverse

 
Graph Inverse

 

Finding the inverse algebraically:


Set the equation equal to y.

Swap the x and y variables.

Utilize the definition of the log function.
(put in exponential form)
 

How to use your
TI-83+/84+ graphing calculator with logarithms.
Click calculator.