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Graph of Logarithm Equation

So what does a log graph look like?

There are two main 'shapes' that a logarithmic graph takes. Depending on whether b in the equation $$ y= log_b (x) $$ is less than 1 or greater than 1.

relationship between equation and graph of logarithm

Note: The two pictures up above do not include the case of b = 1.

Can you figure out why? What is special about the graph of $$ y = log_1 (x) $$?

Properties of Graph

Below you can see the graphs of 3 different logarithms. As you can tell, logarithmic graphs all have a similar shape.

Graphs of logarithms
Property 1

All logarithmic graphs pass through the point.

Property 2

The domain is:

All positive real numbers (not zero).

$$ \{x: x \in \mathbb{R}^+\} $$
Property 3

The range is: all real numbers.

$$ \{x: x \in \mathbb{R}\} $$

Property 4

It is a one-to-one function.

All real numbers. $$ \{x: x \in \mathbb{R}\} $$

Property 5

Has an asymptote that is a vertical line :

Property 6

Grow very slowly for large X ( explore with this applet ).

Property 7

Does not have horizontal asymptotes

Practice Problems

Problem 1

Can you identify which equation below represents a logarithmic equation?

Logarithmic graphs.

graph of logarithm

What are some applications of Logarithmic Graphs?

There are many real world examples of logarithmic relationships. Logarithms graphs are well suited.

  • When you are interested in quantifying relative change instead of absolute difference. Consider for instance the graph below.
  • When you want to compress large scale data. Consider for instance that the scale of the graph below ranges from 1,000 to 100,000 on the y-axis and 1 to 100 on the x-axis--such large scales which can be typical in scientific data are often more easily represented logarithmically.
  • pH scale for measuring acidity of a substance, decibel levels for audio measurement and the Richter scale for measuring earthquakes are all logarithmic. Read more at wikipedia about this.
real world example of logarithmic graph
Back to Logarithms Next to Logarithm Graph Applet