Now that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions. The family of logarithmic functions includes the parent function
along with all its transformations: shifts, stretches, compressions, and reflections.
We begin with the parent function
Because every logarithmic function of this form is the inverse of an exponential function with the form
their graphs will be reflections of each other across the line
To illustrate this, we can observe the relationship between the input and output values of
and its equivalent
in
[link].
Using the inputs and outputs from
[link] , we can build another table to observe the relationship between points on the graphs of the inverse functions
and
See
[link].
As we’d expect, the
x - and
y -coordinates are reversed for the inverse functions.
[link] shows the graph of
and
Observe the following from the graph:
has a
y -intercept at
and
has an
x - intercept at
The domain of
is the same as the range of
The range of
is the same as the domain of
Characteristics of the graph of the parent function,
f (
x ) = log
b (
x )
For any real number
and constant
we can see the following characteristics in the graph of
[link] shows how changing the base
in
can affect the graphs. Observe that the graphs compress vertically as the value of the base increases. (
Note: recall that the function
has base
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