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 $$ y = log_1 (x) $$?

Example

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


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

Property 6

Grow very slowly for large X (see application below)

Property 7

Do have horizontal asymptotes?

Not
Property 8

Based on the table of values below, exponential and logarithmic equations are

logarithms vs exponential function as inverses and table of values
Property 9

As the inverse of an exponential function, the graph of a a logarithm is a reflection across the line y=x of its associated exponential equation's graph.

picture of logarithm graphs as inverses of exponential functions

Practice Problems

Problem 1

Can you identify which equation below represents a logarithmic equation?

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 #1 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
next to Logarithm Graph Applet

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