Logarithms Tutorial

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Logarithms Tutorial

• Understanding the Log Function

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Where Did Logs Come From?

• The invention of logs in the early 1600s fueled
  the scientific revolution. Back then scientists,
  astronomers especially, used to spend huge
  amounts of time crunching numbers on paper.
• By cutting the time they spent doing arithmetic,
  logarithms effectively gave them a longer
  productive life.
  
• The slide rule was nothing more than a device
  built for doing various computations quickly,
  using logarithms.
  

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There are still good reasons for studying them.

• To model many natural processes, particularly in
  living systems. We perceive loudness of sound as
  the logarithm of the actual sound intensity, and
  dB (decibels) are a logarithmic scale.
• To measure the pH or acidity of a chemical
  solution.
  
• To measure earthquake intensity on the Richter
  scale.

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How they are developed

• In the mathematical operation of addition we take
  two numbers and join them to create a third 4
  4 8
  
• We can repeat this operation 4 4 4 12
• Multiplication is the mathematical operation that
  extends this 3 4 12
  
• In the same way, we can repeat multiplication 3
  3 3 27
  
• The extension of multiplication is
  exponentiation 3 3 3 27 33

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More on development

• The exponential function y 2x is shown in this
  graph

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More on development

• Now consider that we have a number and we want to
  know how many 2's must be multiplied together to
  get that number.
  
• For example, given that we are using 2' as the
  base, how many 2's must be multiplied together to
  get 32? That is, we want to solve this equation
  2B 32
• Of course, 25 32, so B 5. To be able to get a
  hold of this, mathematicians made up a new
  function called the logarithm log2 32 5

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Inverses

• This graph was created by switching the x and y
  of the exponential graph, which is the same as
  flipping the curve over on a 45 degree line.
  

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DEFINITION

• The base a logarithm function y logax is the
  inverse of the base a exponential function y ax
  (a 0, a ? 1)

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How to Convert Between Different Bases

• Calculators and computers generally don't
  calculate the logarithm to the base 2, but we can
  use a method to make this easy.
  
• Take for example, the equation 2x 32. We use
  the change of base formula!!  We can change any
  base to a different base any time we want.  The
  most used bases are obviously base 10 and base e
  because they are the only bases that appear on
  your calculator!

Pick a new base and the formula says it is equal
to the log of the number in the new base
divided by the log of the old base in the new
base.
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Examples

• Find the value of log2 37
• Change to base 10 and use your calculator. log
  37/log 2
  
• Now use your calculator and round to hundredths.
  
  
• 5.21
• Log7 99 ?
• Change to base 10 or base e.  Try it both ways
  and see.
  
• log3 81
• log4 256
• log2 1024

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Properties Of Logarithms

• For any real numbers x 0 and y 0,
• ? Product Rule loga xy loga x loga y
• ? Quotient Rule
• ? Power Rule loga xy yloga x

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More Practice

• Express each as a single log.
• Log x Log y - Log z
• 2 Ln x 3 Ln y
• Solve
• Log2 (x 1) Log2 3 4
• Log (x 3) Log x 1

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Web Sites

• http//www.shodor.org/UNChem/math/logs/
• http//www.physics.uoguelph.ca/tutorials/LOG/
• http//www.purplemath.com/modules/logs.htm
• http//www.exploremath.com/activities/Activity_pag
  e.cfm?ActivityID7
  
• SAMPLE Test on Logs
• http//www.alltel.net/okrebs/page58.html