Section 4.4Logarithmic Scales

Plot the distances on a number lineObject Distance (million km)

Mercury 58

Venus 108

Earth 149

Mars 228

Jupiter 778

Saturn 1426

Uranus 2869

Neptune 4495

Pluto 5900

Proxima Centauri 4.1 x 107

Andromeda Galaxy 2.4 x 1013

• If you haven’t already, try plotting the data on an axis labeled 100, 101, 102, 103, and so on– How do you decide where to put each planet on

this scale?– Can we fit all the planets now?

• In order to fit all the planets on this scale you must convert their distances to powers of 10– How do we do this?– Recall

• What we have created is a logarithmic scale

xxy y 10thenlogIf

Use of logs to fit an exponential function to data

• Consider the following table of data

• Create a scatter plot with your calculator• What form does the data have?

x y

1 210

2 405

3 833

4 1590

5 3245

6 6510

• We can use logarithms to make this data linear

• Make a scatter plot of x vs. ln(y)• How does the data look?• Find both the exponential and linear regression

functions

x y ln(y)

1 210 5.34

2 405 6

3 833 6.72

4 1590 7.37

5 3245 8.08

6 6510 8.78

• In your groups try problems 12 and 15