Logarithmic Scales

1
Logarithmic Scales

• Lesson 4.4A

2
Graphing Difficult Data

• Some data can be easily considered on a linear
  scale 
• Weights of a team of football players
• Heights of a team of volleyball players
• GPA's of a class of students

3
Graphing Difficult Data

• Other things are more difficult to represent on a
  linear graph due to the vast range of values 
• There are data of several orders of magnitude
• Examples
• distance to the moon, to planets
• light brightness
• loudness of a sound

4
How to Graph These Numbers?

• Consider thevast rangeof the numbers

Distance from the Sun Distance from the Sun
Object Distance (million km)
Mercury 58
Venus 108
Earth 149
Mars 228
Jupiter 778
Saturn 1426
Uranus 2869
Neptune 4495
Pluto 5900
Proxima Centauri 4.1E07
Andromeda Galaxy 2.4E13
5
How to Graph These Numbers?

• What's wrongwith thispicture?

6
How to Graph These Numbers?

• What's wrongwith thispicture?

We need a way toset a scale that fits all the
data
7
How to Graph These Numbers?

• The solutionSet the scaleto be the
  exponentof the distance

This is called a logarithmic scale
8
 Using Exponents to Indicate Distance

• Consider the following problem 
• Given a number y ...   what value of x exists
  such that 

x y
-2 .. -1 0.01 .. 0.1
 -1 .. 0 0.1 .. 1
0 .. 1 1 .. 10
1 .. 2 10 .. 100
9
 Using Exponents to Indicate Distance

• Note the value we need to represent for Mercury's
  distance from the sun at 58 million kilometers,

n, exponent 1.5 1.55 1.6 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00
10n 31.62 35.48 39.81 44.67 50.12 56.23 63.10 70.79 79.43 893.1 100
Somewhere between these two
For Mercury at 58 million km  58 101.76343
10
Assignment

• Lesson 4.4A
• Page 181
• Exercises
• 1 5, 9, 11, 13, 15