# 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 GraphExamples

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).

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.

### 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).

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.

## Relationship with the Exponential Function

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.

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