Write

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Write

• What is a logarithm?
• What is it the opposite of?
• What is the value of log6216?
• Why?
• Make this statement of mathematical fact without
  using logs.
• What do we call that little subscript number
  there?

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Review 8Logarithms and Solving

• Thursday 1 June 2006

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Exponents
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A New Idea The Logarithm
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Logs have a base

• They ask the question, What power do I have to
  raise the base to to get the number in question?

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What are commonly-used bases?

• Well, 10, of course.
• Why? For the same reason that we have 10 digits
  namely, that we have 10 digits!
• This is called the common log, and is written as
  just plain log.
• Also, e (2.71828)
• Why? This is slightly more complex. Well get
  to it.
• This is called the natural logarithm, and is
  written ln.
• So log log10, and ln loge.

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Logarithms are about notation.

• One statement you might make about the numbers 3,
  4, and 81 is this
• 3 to the 4th is 81.
• 3481
• Logs give us another way to express this
  relationship
• Log base 3 of 81 is 4.
• Log3814
• We can find the value of a log using the
  calculator.

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Equivalent Statements

• Exponential Function
• 22 4
• 32 9
• 42 16
• 53 125
• 63 216
• (1/3)3 1/27
• 2-2 1/4

• Logarithmic Function
• log2 4 2
• log3 9 2
• log4 16 2
• log5 125 3
• log6 216 3
• log1/3 1/27 3
• log2 ¼ -2

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Logarithms are functions.

• Logs are the inverses of exponents.
• That is, if the function is
• f(x) 7x
• The inverse is
• g(x) log7x

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A couple of examples
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Log rules are like exponent rules.

• Exponent multiplication rule
• xa xb xab
• Exponent division rule
• xa xb xa-b
• Exponent exponent rule
• (xa )b xab

• Log multiplication rule
• logx(ab) logxa logxb
• Log division rule
• logx(a/b) logxa logxb
• Log exponent rule
• logxab blogxa

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Simplify

• log 4 2 log 5
• 3 log4 8 2 log4 2
• 2 log 10 log 30

• log7 75
• 2x3 log2x4
• 2 log 10 2x-9

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Change of Base Formula
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Well, so what?

• Like lots of other skills we learn in math
  factoring, simplifying, combining like terms
  the ability to deal with logarithms is about
  flexibility.
• Sometimes, theyre part of the problem from the
  beginning.
• Sometimes, introducing them as a method of
  representation is useful.

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Consider

• Solve for x
• 92x-7 27x-3
• Our options are limited.
• Last week, all wed have had was guess-and-check.
• But now, check out what happens if we take the
  base-3 log of both sides.

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Did you get that?

• When you look at
• 2x 4 16
• you say to yourself, Well, Ive got to get that
  x alone, so Ill subtract 4 and then divide by
  2.

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Logs Help Us Solve

• Logs can do the same trick they can help us deal
  with the fact that the x isnt isolated, so we
  can solve
• Division gets rid of coefficients.
• Addition gets rid of constant terms.
• Roots can get rid of exponents.
• Logs help us bring variables in exponent
  positions down to where we can deal with them!

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Two Examples

• 4x 32x

• 52x 252x-4

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Practice

• Simplify into a single equation, then solve for x
  in each of these equations.
• log2(3x1) 5
• log 2x log x 50
• 53x 500
• 642x1 164x
• 113x 501x-6

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For Example
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Steps

• Apply a log to both side.
• Apply the log exponent rule to turn exponents
  into factors.
• Compute the value of the log.
• Distribute.
• Solve.

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A More Complicated Example
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How do we choose a base?

• On the one hand, you might always use 10, because
  thats whats on your calculator.
• Here, youll be estimating, so use several
  decimals.
• On the other, you could look for common factors.
• This requires some brain.
• Its also not always possible.

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Practice No Calculators!!!

• Simplify into an equation with no exponents, then
  solve for x in each of these equations.

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Practice Calculators OK!!!