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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.

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

Property 1

All logarithmic graphs pass through the point.

(1,0)
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

##### Property 8

Remember: Inverse functions have 'swapped' x,y pairs.

### Practice Problems

##### Problem 1

Logarithmic graphs.

#### 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.