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Logarithmic Function Graph Examples


The exponential function and the logarithmic function and their relationship were featured on the exponential function and logarithms introduction page.


When dealing with real numbers.

Logarithmic Function.     f(x) \space {\small{=}} \space{\tt{log}}_a (x)   ,     a > 0 \space \space , \space \space a \space {\cancel{=}} \space 1 \space \space , \space \space x > 0.

Exponential function.     f(x) = a^x,     a > 0


The functions are inverses of each other.

{\tt{log}}_a (a^x) \space {\small{=}} \space x         =>         a^{{\tt{log}}_a (x)} \space {\small{=}} \space x



Here we will look at some logarithmic function graph examples. Observing the standard form of the graphs, and how they differ on a cartesian axis.






Logarithmic Function Graph
Examples



When we have,    f(x) \space {\small{=}} \space{\tt{log}}_a (x)   ,     a > 0 \space \space , \space \space a \space {\cancel{=}} \space 1 \space \space , \space \space x > 0.

The shape of the graph will be influenced by  a.



a > 1

Let’s firstly look at the case of,    a > 1.

f(x) \space {\small{=}} \space {\tt{log}}_2 (x)


We can obtain some points on the graph.

f(\frac{1}{4}) \space {\small{=}} \space {\tt{log}}_2 (\frac{1}{4}) \space {\small{=}} \space {\text{-}}2 \space \space \space \space \space \space \space \space {\footnotesize{( 2^{{\text{-}}2} = \frac{1}{4} )}}

f(\frac{1}{2}) \space {\small{=}} \space {\tt{log}}_2 (\frac{1}{2}) \space {\small{=}} \space {\text{-}}1 \space \space \space \space \space \space \space \space {\footnotesize{( 2^{{\text{-}}1} = \frac{1}{2} )}}

f(1) \space {\small{=}} \space {\tt{log}}_2 (1) \space {\small{=}} \space 0 \space \space \space \space \space \space \space \space \space \space {\footnotesize{( 2^0 = 1 )}}

f(2) \space {\small{=}} \space {\tt{log}}_2 (2) \space {\small{=}} \space 1 \space \space \space \space \space \space \space \space \space \space {\footnotesize{( 2^1 = 2 )}}

f(4) \space {\small{=}} \space {\tt{log}}_2 (4) \space {\small{=}} \space 2 \space \space \space \space \space \space \space \space \space \space {\footnotesize{( 2^2 = 4 )}}


So we have found the points   (\frac{1}{4},{\text{-}}2) , (\frac{1}{2},{\text{-}}1) , (1,0) , (2,1) , (4,2).


This information can now help in drawing the graph for   f(x) \space {\small{=}} \space {\tt{log}}_2 (x).

First image for the Logarithmic Function Graph Examples page.


This logarithm graph above is the standard shape for a function   f(x) \space {\small{=}} \space {\tt{log}}_a (x)   where   a > 1.

We can observe the following.

  As  x  gets smaller towards 0, the graph trends to negative infinity.
  As  x  gets larger, the graph trends to positive infinity.
  The graph is increasing and doesn’t cross the  y-axis.
  The graph will pass through  (1,0),  and for  f(x) \space {\small{=}} \space {\tt{log}}_a (x),  will pass through  (a,1).



Here is another diagram with 2 other log function graphs for   a > 1.
Where we can see how the shape is affected by a varying value of  a.


Two Logarithm Graphs where the base is greater than 1.





a < 0 < 1

The other case for logarithm graphs is,    0 < a < 1.

We can examine    f(x) \space {\small{=}} \space {\tt{log}}_{\frac{1}{2}} (x).


Again obtaining some points on the graph.

f(\frac{1}{4}) \space {\small{=}} \space {\tt{log}}_{\frac{1}{2}} (\frac{1}{4}) \space {\small{=}} \space 2 \space \space \space \space \space \space \space \space \space \space {\footnotesize{( \space (\frac{1}{2})^2 = \frac{1}{4} \space )}}

f(\frac{1}{2}) \space {\small{=}} \space {\tt{log}}_{\frac{1}{2}} (\frac{1}{2}) \space {\small{=}} \space 1 \space \space \space \space \space \space \space \space \space \space {\footnotesize{( \space (\frac{1}{2})^1 = \frac{1}{2} \space )}}

f(1) \space {\small{=}} \space {\tt{log}}_{\frac{1}{2}} (1) \space {\small{=}} \space 0 \space \space \space \space \space \space \space \space \space \space \space {\footnotesize{( \space (\frac{1}{2})^0 = 1 \space )}}

f(2) \space {\small{=}} \space {\tt{log}}_{\frac{1}{2}} (2) \space {\small{=}} \space {\text{-}}1 \space \space \space \space \space \space \space \space \space \space {\footnotesize{( \space (\frac{1}{2})^{{\text{-}}1} = 2 \space )}}

f(4) \space {\small{=}} \space {\tt{log}}_{\frac{1}{2}} (4) \space {\small{=}} \space {\text{-}}2 \space \space \space \space \space \space \space \space \space \space {\footnotesize{( \space (\frac{1}{2})^{{\text{-}}2} = 4 \space )}}


We have found the points   (\frac{1}{4},2) , (\frac{1}{2},1) , (1,0) , (2,{\text{-}}1) , (4,{\text{-}}2).

This information can now help in drawing the graph for   f(x) \space {\small{=}} \space {\tt{log}}_{\frac{1}{2}} (x).

Logarithm Graph with base a less than 1.


This logarithm graph above is the standard shape for a function   f(x) \space {\small{=}} \space {\tt{log}}_a (x)   where   0 < a < 1.

The following can be observed.

  As  x  gets smaller towards 0, the graph trends to positive infinity.
  As  x  gets larger, the graph trends to negative infinity.
  The graph is decreasing and doesn’t cross the  y-axis.
  The graph will pass through  (1,0),  and for  f(x) \space {\small{=}} \space {\tt{log}}_a (x),  will pass through  (a,1).



Like before, we can view another diagram with 2 other log function graphs for   0 < a < 1.
Observing how the graph shape is affected by a varying value of  a.


Two Log Graphs where the base is less than 1.






Relationship with the Exponential Function Graph


A logarithmic function is the inverse of its corresponding exponential function.

As such, the graph of the functions are reflected in the line  y = x.

The coordinates get swapped around.

Logarithmic and Exponential function graphs on same axis, reflected in the line y=x.






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