The Logarithmic Function

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The Logarithmic Function

• Lesson 5.3

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Why?

• What happens when you enter into your calculator
• If we want to know about limitations on the
  domain and range of the log function

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Graph, Domain, Range

• Use your calculator to discover facts about the
  log function
• In the Y screen, specify log(x)
• Set tables with ?T initial x 0, ?x
  0.1
• View the tables

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Graph, Domain, Range

• Note domain for 0 lt x lt 1
• Change the ?x to 5, view again

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Graph, Domain, Range

• View graph with window -1 lt x lt 10, -4 lt y lt 5
• Why does thegraph appearundefinedfor x lt 0 ?

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Graph, Domain, Range

• Recall that
• There can be no value for y that gives x lt 0
• Domain for y log x
• x gt 0
• Range
• y all real values

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Vertical Asymptote

• Note behavior of function as x ? 0

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Inverse Functions

• Recall use of the DrawInv command on the graph
  screen

You type in y1(x)
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Inverse Functions

• Now consider the functionsy ln x and y
  ex
• Place in Y screen
• Specify zoom standard, then zoom square
• Note relationship of the two functions
• Graph y x on same graph
• Graphs are symmetricabout y x
• Shows they are inverses

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Assignment

• Lesson 5.3A
• Page 203
• Exercises
• 1 13 odd, 19 21 all

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Usefulness of Logarithms

• Logarithms useful in measuring quantities which
  vary widely
• Acidity (pH) of a solution
• Sound (decibels)
• Earthquakes (Richter scale)

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Chemical Acidity

• pH defined as pH -logH
• where H is hydrogen ion concentration
• measured in moles per liter
• If seawater is H 1.110-8
• then log(1.110-8) 7.96

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Chemical Acidity

• What would be the hydrogen ion concentration of
  vinegar with pH 3?

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Logarithms and Orders of Magnitude

• Consider increase of CDs on campus since 1990
• Suppose there were 1000 on campus in 1990
• Now there are 100,000 on campus
• The log of the ratio is the change in the order
  of magnitude

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Logarithms and Orders of Magnitude

• We use the log function because it counts the
  number of powers of 10
• This is necessary because of the vast range of
  some physical quantities we must measure
• Sound intensity
• Earthquake intensity

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Decibels

• Suppose I0 is the softest sound the human ear
  can hear
• measured in watts/cm2
• And I is the watts/cm2 of a given sound
• Then the decibels of the sound is

The log of the ratio
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Decibels
Approx. Decibel Level Example
0 Faintest sound heard by human ear.
30 Whisper, quiet library.
60 Normal conversation, sewing machine, typewriter.
90 Lawnmower, shop tools, truck traffic 8 hours per day is the maximum exposure to protect 90 of people.
100 Chainsaw, pneumatic drill, snowmobile 2 hours per day is the maximum exposure without protection.
115 Sandblasting, loud rock concert, auto horn 15 minutes per day is the maximum exposure without protection.
140 Gun muzzle blast, jet engine noise causes pain and even brief exposure injures unprotected ears. Maximum allowed noise with hearing protectors.
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Decibels

• If a sound doubles, how many units does its
  decibel rating increase?
• Find out about hearing protection
• How many decibels does it reduce the sound
• How much does that decrease the intensity of the
  sound?

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Measuring Earthquakes

• S-wave
• Surface-wave
• P-wave
• Pressure-wave

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Measuring Earthquakes
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Measuring Earthquakes

• Seismic waves radiated by all earthquakes can
  provide good estimates of their magnitudes

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Definition of Richter Scale

• Magnitude of an earthquake with seismic waves of
  size W defined as
• We measure a given earthquake relative to the
  strength of a "standard" earthquake

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Comparable Magnitudes

• Richter TNT for Seismic Example
• Magnitude Energy Yield (approximate)
• -1.5 6 ounces Breaking a rock on
  a lab table
• 1.0 30 pounds Large Blast at a
  Construction Site
• 1.5 320 pounds
• 2.0 1 ton Large Quarry or
  Mine Blast
• 2.5 4.6 tons
• 3.0 29 tons
• 3.5 73 tons
• 4.0 1,000 tons Small Nuclear
  Weapon
• 4.5 5,100 tons Average Tornado
  (total energy)
• 5.0 32,000 tons
• 5.5 80,000 tons Little Skull Mtn.,
  NV Quake, 1992
• 6.0 1 million tons Double Spring
  Flat, NV Quake, 1994
• 6.5 5 million tons Northridge, CA
  Quake, 1994
• 7.0 32 million tons Hyogo-Ken Nanbu,
  Japan Quake, 1995
  Largest Thermonuclear Weapon
• 7.5 160 million tons Landers, CA Quake,
  1992
• 8.0 1 billion tons San Francisco, CA
  Quake, 1906

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Assignment

• Lesson 5.3B
• Page 204
• Exercises
• 14 18 all, 23 33 Odd