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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), and Log

(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)

Data plotted with linear scale

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)

Plot using Logs

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.

Also make sure you can use it to generate this

table.

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 102100.
- 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.

Also make sure you can use it to generate this

table.

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.

This is the End ...

Return to Math Tutorial Page Return to Prof.

Jackels Home Page

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