﻿ Graph of Logarithm: Properties, example, appearance, real world application, interactive applet

# 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 $$y = log_1 (x)$$?

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

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

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 ### Ultimate Math Solver (Free)

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