LOGARITHMIC FUNCTION

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LOGARITHMIC FUNCTION

• Jasna KOS
• Gimnazija Bežigrad, Ljubljana
• Matija LOKAR
• Faculty of mathematics and physic
• University of Ljubljana, SLOVENIA

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Slovene school system

• Various types of secondary schools
• Gymnasium (grammar school)
• 15 years
• 4 years
• Preparation for higher education (University)
• Ends with Matura central external examination

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Logarithmic function

• Second year of gymnasium
• 10th year of schooling (soon 11th)
• 16 year old students
• Inverse function of exponential function

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Approaches to teaching

• Different materials, styles
• Classical lecturing/demonstrations/lab work
• Videos
• Computer prepared lectures
• Internet
• Self discovering
• Active participation

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Workshop

• Introduction to the materials
• The Rules of Logarithmic Computation
• Translation and Scaling
• Solutions to Equations and Inequalities
• Using logarithmic function
• In Psychology
• Music
• Loudness of Sound
• Fractals

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Workshop

• WWW
• http//rc.fmf.uni-lj.si/matija/logarithm/logfun.ht
  m
• Brief explanation of all worksheets
• Introduction to two worksheets (with necessary
  DERIVE command explained)
• Translation and scaling
• Using logarithmic function in psychology
• Group work choose a worksheet
• ? Discussion

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Worksheets

• Teachers part
• explanations,
• what to expect,
• sometimes solutions, hints
• Attention to technical aspects
• Students part
• Completely prepared worksheets
• Teachers are encouraged to use prepared work only
  as a basis

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The rules of logarithmic computation

• http//rc.fmf.uni-lj.si/matija/logarithm/worksheet
  s/rules.htm
• http//rc.fmf.uni-lj.si/matija/logarithm/teacher/r
  ules.htm
• Discovering three main rules
• Addition, subtraction, powers
• Lab exercise or as a demonstration
• Quite short exercise
• discussion
• WEB

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The rules of logarithmic computation

• On history of logarithmic function
• http//www-history.mcs.st-and.ac.uk/history/Mathem
  aticians/Napier.html
• http//www.sosmath.com/algebra/logs/log1/log1.html
  
• http//britannica.com/bcom/eb/article/8/0,5716,118
  1783,00.html
• http//www-history.mcs.st-and.ac.uk/history/Mathem
  aticians/Briggs.html

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The rules of logarithmic computation

• On Eulers' number e and natural logarithm
• http//mathforum.com/dr.math/faq/faq.e.html
• http//www.math.utoronto.ca/mathnet/answers/ereal.
  html
• http//duke.usask.ca/fowlerr/e.html

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The rules of logarithmic computation

• On rules for calculating with logarithms
• http//www.sosmath.com/algebra/logs/log4/log41/log
  41.html
• http//www.shodor.org/UNChem/math/logs/index.html
  
• http//www.physics.uoguelph.ca/tutorials/LOG/index
  .html
• http//www.math.utah.edu/alfeld/math/log.html
• http//www.cne.gmu.edu/modules/dau/algebra/exponen
  ts/lexercises_frm.html
• http//taipan.nmsu.edu/aght/soils/soil_physics/tut
  orials/log/log_home.html
•  and many more, especially on Ask Dr. Math
•   http//forum.swarthmore.edu/dr.math/tocs/logarit
  hm.high.html

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The rules of logarithmic computation

• Log(x) in DERIVE denotes natural logarithm
• Log(x, 10) is a common logarithm
• Natural logarithms are used
• Be careful about settings in DERIVE
• No difficulties with math. notation of rules
  but expressing rules with words?
• Next lesson
• Summarizing the findings
• Proofs

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Translation and scaling

• Families of functions
• Ln(x a)
• a Log5(x)
• Loga(x)
• How a logb(x c) looks like
• Next hour summarize the findings curriculum
  requires capability of drawing log functions by
  hand

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Translation and scaling

• Students are already familiar with the basic
  graph of logarithmic function
• Brief explanation on DERIVE use
• Log(x) is not a common logarithm

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Log(x) ln(x)
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Log(x) ln(x)
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Log(x) ln(x)
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Log(x) ln(x)
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Translation and scaling

• Different settings
• Observation of the teachers presentation
• Fill in the worksheets
• Home work

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Solutions to equations and inequalities

• Most present exercises elementarily solvable
• Most harder exercises marked and avoided
• Graphic approach
• Numerical solution with DERIVE

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Using logarithmic function

• Trying to avoid classical examples
• Present examples
• Music (musical ladders)
• Sound (loudness)
• Curve of forgetting (psychology)
• Fractals
• Additional suggestions on WWW

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Using logarithmic function

• Rocket equations (http//www.execpc.com/culp/rock
  ets/rckt_eqn.html) with calculating how high the
  rocket model will go (http//www.execpc.com/culp/
  rockets/qref.html)
• Exercises in Math Readiness on http//math.usask.c
  a/readin/menu.html with number of exercises where
  log function is needed http//math.usask.ca/readi
  n/examples/expgrtheg.html

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Using logarithmic function

• Radioactive decay
• http//math.usask.ca/readin/examples/expdeceg.html
  
• On earthquakes and Richter Magnitude
  http//www.seismo.unr.edu/ftp/pub/louie/class/100/
  magnitude.html
• Many different problems http//forum.swarthmore.e
  du/dr.math/tocs/logarithm.high.html

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Curve of forgetting

• F(t) A B log(t 1)
• A, B experimentally determined
• Simple math model
• Weakness of the model
• Still some results possible
• We show why logarithmic function is needed
• Why is it necessary to interpret the results
  (calculation when we forget everything)

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Mathematics and music

• Frequencies, hearing, C-minor scale, octaves
• Table of frequencies for the tones in the first
  octave
• Regression curve
• Due to interest of the students suitable utility
  function
• Introduction of the Least squares method

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Loudness of sound

• Logarithmic scale
• Connection between the intensity and loudness
• Real life examples
• Different questions about the noise
• Can we add loudness (noise)?

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Fractals

• Students hear about them in art, nature, ...
• Mathematic concepts necessary
• Fractal fractioned dimension
• Homework, cauliflower,
• Dimension of a fractal
• Introduction to programming (recursion, ...)

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SAMPLE Translation and Scaling
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SAMPLE Curve of forgetting
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YOUR WORK