Logarithms

Tutorial to explain the nature of logarithms and their use in our

courses.

What is a Logarithm?

• The common or base-10 logarithm of a number is the power to which 10 must be raised to give the number.

• Since 100 = 102, the logarithm of 100 is equal to 2. This is written as:

Log(100) = 2.

• 1,000,000 = 106 (one million), andLog (1,000,000) = 6.

Logs of small numbers

• 0.0001 = 10-4, and Log(0.0001) = -4.All numbers less than one have negative logarithms.

• As the numbers get smaller and smaller, their logs approach negative infinity.

• The logarithm is not defined for negative numbers.

Numbers not exact powers of 10

• Logarithms are defined for all positive numbers.

• Since Log (100) = 2 and Log (1000) = 3, then it follows that the logarithm of 500 must be between 2 and 3.

• In fact, Log(500) = 2.699

Small Numbers not exact powers of 10

• Log(0.001) = -3 and Log (0.0001) = - 4

• What would be the logarithm of 0.0007?Since it is between the two numbers above, its logarithm should be between -3 and -4.

• In fact, Log (0.0007) = -3.155

Why Logarithms?• In scientific applications it is common to compare

numbers of greatly varying magnitude. Direct comparison of these numbers can be difficult. Comparison by order of magnitude using logs is much more effective.

• Time scales can vary from fractions of a second to billions of years.

• You might want to compare masses that vary from the mass of an electron to that of a star.

• The following table presents an example:

Years before present (YBP)

Formation of Earth 4.6 x 109 YBP

Dinosaur extinction 6.5 x 107 YBP

First hominids 2 x 106 YBP

Last great ice age 1 x 104 YBP

First irrigation of crops 6 x 103 YBP

Declaration of Independence 2 x 102 YBP

Establishment of UWB 1 x 10 YBP

Data plotted with linear scaleEvents from Table I

0.E+00

1.E+09

2.E+09

3.E+09

4.E+09

5.E+09

Earth

Dinosa

urs

Homini

ds

Ice

Age

Irrig

atio

n

Inde

pend

ence

UWB

Yea

rs b

efo

re p

rese

nt

All except the first two data points are hidden on the axis.

Use Logs of Ages

• Because the data spans such a large range, the display of it with a linear axis is useless. It makes all events more recent than the dinosaurs to appear the same!

• Instead, plot the logarithm of the tabular data. Now the range to be plotted will be much smaller, and the plot will distinguish between the ages of the various events.

Log (YBP)EVENT YBP Log(YBP)

Formation of Earth 4.6 x 109 9.663

Dinosaur extinction 6.5 x 107 7.813

First hominids 2 x 106 6.301

Last great ice age 1 x 104 4.000

First irrigation of crops 6 x 103 3.778

Declaration of Independence 2 x 102 2.301

Establishment of UWB 1 x 10 1.000

Plot using Logs

Events from Table I

0

2

4

6

8

10

Earth

Dinosa

urs

Homini

ds

Ice

Age

Irrig

atio

n

Inde

pend

ence

UWB

Lo

g(Y

BP

)

All data are well represented despite their wide range.

Your calculator should have a button marked LOG. Make sure you can use it to generate

this table.

N N as power of 10 Log (N)

1000 103 3.000

200 102.301 2.301

75 101.875 1.875

10 101 1.000

5 100.699 0.699

Also make sure you can use it to generate this table.

N N as power of 10 Log (N)

1 100 0

0.1 10-1 -1

0.062 10-1.208 -1.208

0.001 10-3 -3

0.00004 10-4.398 -4.398

Antilogs?

• The operation that is the logical reverse of taking a logarithm is called taking the antilogarithm of a number. The antilog of a number is the result obtained when you raise 10 to that number.

• The antilog of 2 is 100 because 102=100.• The antilog of -4 is 0.0001 because 10-4 = 0.0001

Find the antilog function on your calculator.

• To take antilogs, your calculator should have one of the following:

• A button marked LOG-1

• A button marked 10x

• A button marked ALOG

• A two-button sequence such as INV followed by LOG.

Make sure you can use your calculator to generate this table.

N As a power of 10 Antilog(N)

3 103 1000

1.5 101.5 31.62

1 101 10

0 100 1

-2 10-2 0.01

-3.4 10-3.4 0.0003981

Also make sure you can use it to generate this table.

N N as power of 10 Log (N)

1 100 0

0.1 10-1 -1

0.062 10-1.208 -1.208

0.001 10-3 -3

0.00004 10-4.398 -4.398

Natural Logarithms

Some calculators (especially business models) may have only natural logarithms. These can be used to obtain common (base-10) logarithms and antilogs. See the tutorial on Natural Logs if this the case for you.

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