Logarithm Laws
Turning Multiplication into AdditionFurther LawsChange of BaseExponential and Logarithm GraphsLogarithm Scales and Slide Rules

Exponential and Logarithm Graphs

Here is a table that you might use to plot the graph of .

x
-3
-2
-1
0
1
2
3
4
5
6
1
2
4
8
16
32
64

and here is one that you could use for

x
1
2
4
8
16
32
64
-3
-2
-1
0
1
2
3
4
5
6

You do not need to calculate values for the logarithm graph. You just use the exponential values backwards. In fact if you get really lazy you don't even have to draw a new graph. Here is the graph of that we used earlier but from a different perspective.

I've flipped it over and it is now a perfect logarithm graph. Sure the numbers on the axes look funny but don't get too fussy. If you would like to do this flip yourself just go to the Pop up Graph.

Notice that the exponential graph only took positive values. This means you can only put positive values into a logarithm. Logarithms are not defined for negative numbers.

Symmetry of inverse functions.

Whever two operations undo each other their graphs will be perfectly symmetrical, This is the case for exponential and logarithm graphs. The symmetry is evident in the table above. It is also clear when we draw both graphs on the same axes. Note that to see the symmetry geometrically it is necessary to have the same scale on both axes.

The dynamic graphs can be used to investigate any of the following.

  • the symmetry in all the different types of logarithm and exponential graphs.
  • the effect of changing the base on both graphs.
  • the monotonic nature of both graphs (move the point)
  • pay attention to the slopes at the (0,1) and (1,0) points - for what value of the base are the slopes here equal to one (that is the tangents are parallel to the line of symmetry and to each other)?
Turning Multiplication into AdditionFurther LawsChange of BaseExponential and Logarithm GraphsLogarithm Scales and Slide Rules
 
 
 
 
 
 
 


 


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