Exponential Functions: Review

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Exponential Functions Review

• Viscosity of fluid (or mantle rocks) is
  strongly effected by water content.
  
• A little goes a long way!

? e -T

• What does this look like ?

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Progressing Geometrically

• Arithmetic Progressions
• Geometric Progressions
• Relating these Progressions
• Logarithms

John Napier (1550-1617)
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The Discovery of Logarithms
John Napier, a Scottish mathematician, physicist,
astronomer, astrologer was trying to find an
easier way to do multiplication and division by
methods of addition and subtraction. He felt his
most important book used methods of Euclidean
deduction to argue that the Pope was the
Antichrist (1593). Here he also predicted the end
of the world in 1786. Later he worked 20 yrs to
develop logarithms, but he was not thinking of
exponents....
John Napier (1550-1617)
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The Discovery of Logarithms

• Mirifici Logarithmorum Canonis
  Descriptio(Description of the Wonderful Canon of
  Logarithms)
  -1614, 1619
• Logarithms revolutionized calculations.
• This was developed at a time when zero was
  not well established. Decimal fractions
• had also not been invented.
• Napier had no concept of exponents!
• Examples like x3 were not invented until 1637
  by Descartes' La geometrie.

• 1748, Euler first published the concept of
  logarithms as exponents.

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The Discovery of Logarithms

• What was Napier doing with logarithms if it had
  nothing to do with exponents ?
• So What ?
• Are logarithms just a mathematical concept, or
  can they help you be a more skilled geologist ?

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The Discovery of Logarithms

• Napier developed logarithms to bridge between
  arithmetic progressions and geometric
  progressions.
• Understanding this connection provides a
  powerful way of looking at semilog and
  log-log graphs.

Search geometric progression...pics
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Progressions, Series, and Sequences
Search pics of sequance, series....
Sequence Succession of numbers which have a
rule for relating one to the next. (terms
separated by commas) - geometric progression -
arithmetic progression Series Terms of a
sequence are added together (terms connected y
plus or minus.
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Arithmetic Progression (AP)
In an arithmetric progression, to go from one
term to the next you just add a number to the
last term. This number is known as the common
difference. In the example below, the common
difference is d.
b, bd, b2d, b3d, b4d, ....
This can also be written as b nd .
(where n starts with n 0)
Example Assume b 0, and d 1 The
progression is
0, 1, 2, 3, 4,
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Arithmetic Progression
What is the common difference value, d, for the
arithmetic progression below ?
5, 8, 11, 14, 17

• To decipher this, assume b nd
  for an arithmetic
  progression.
  
• First term Find b when d 0.
  
  
• Second term What do you add to the first term
  to get the second
  term ?
  
• This gives you the common difference, d.

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Geometric Progression (GP)
In a geometric progression, to go from one term
to the next you just multiply the last term by a
given value known as the common ratio. In the
example below, the common ratio is r.
a, ar, ar2, ar3, ar4, ....
This can also be written as arn .
(where n starts with n 0)
Example Assume a 1, and r 2 The
progression is
1, 2, 4, 8, 16
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Geometric Progression
What is the common ratio for the geometric
progression below ?
5, 15, 45, 135, 405

• To decipher this, assume arn for a geometric
  progression.
• First term Find a when n 0.
  
  
• Second term What do you multiply the first
  term by to get the
  second term ?
  
• This gives you the common ratio, r.

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Can we Relate One Progression to Another ?

• Consider the two progressions below which
  represent counting the population of forams
  for a each species.
• The arithmetic progression (AP) gives years
  from initiation of population counting.
• The geometric progression (GP) gives the number
  of individuals of a particular species (in
  thousands)

AP 0 1 2 3 4 5 6 GP
1 2 4 8 16 32 64
What is the common difference for the arithmetic
progression ? What is the common ratio for
geometric progression ?
This means the population of forams doubles (2)
every year (1).
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Grain Size of Sedimentary Particles

• Pairing arithmetic and geometric progressions
  can be used to study grain size of sediments.
• Wentworth Scale particle sizes vary from clay
  size (lt 1/256 mm)
• to boulder size (diameter gt 256 mm). Sand sizes
  1/16 mm to 2 mm.
• These size increments are follow the powers of 2.
  
• Krumbein (1936) defined grain sizes (f) as

f -log2 (Dmm)
(Here Dmm is the grain diameter but it must
dimensionless and stands for the number of
times longer than a millimeter.)
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How Do We use Logarithms Today ?
Although we do not need to use logarithms to help
us with division and multiplication today, the
relationship between logarithms and exponentials
greatly simplifies many calculations that we use
in geology.
GP 16 8 4 2 1 AP 0
5710 11400 17100 22800
GP is the radioactivity of a fossil measured on a
Geiger counter. AP is the age of the sample. This
describes the decay constant against the age of
the sample.