Logarithms

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Logarithms

• Our Inverted Exponential Friends

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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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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

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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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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.
• 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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Simplify

• log 4 2 log 5
• 3 log4 8 2 log4 2
• 2 ln e ln 3e

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

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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 16
• you say to yourself, Well, Ive got to get that
  x alone, so Ill divide by 2.
• 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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Practice

• Solve for x in each of these equations.
• log(3x1) 5
• log 2x log x 11
• 53x 500
• 642x1 164x

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Summary

• Logarithms are a different way of notating
  exponent relationships.
• They have a base, and a power.
• They can be shoved around using multiplication,
  division, and power rules that look a lot like
  the ones that work for exponents.
• Theyre useful because they can undo
  exponential functions by bringing the exponent
  down to where we can deal with it.