Logarithms

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Logarithms

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

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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.

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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.

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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

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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

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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

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Years before present (YBP)
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Data plotted with linear scale
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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.

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Log (YBP)
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Plot using Logs
All data are well represented despite their wide
range.
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Your calculator should have a button marked LOG.
Make sure you can use it to generate this table.
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Also make sure you can use it to generate this
table.
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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

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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.

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