Lesson 3.3, page 400 Properties of Logarithms

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Lesson 3.3, page 400Properties of Logarithms

• Objective To learn and apply the properties of
  logarithms.

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Real-World Connection

• Logarithms are used in applications involving
  sound intensity decibel level.

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Think about this

• If a logarithm is the inverse of an exponential,
  what do you think we can surmise about the
  properties of logarithms?
• They should be the inverse of the properties of
  exponents! For example, if we add exponents when
  we multiply in the same base, what would we do to
  logs when they are being multiplied?

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PRODUCT RULE, page 400

• Product Property logb(MN) logbM logbN
• The logarithm of a product is the sum of the
  logarithms of the factors.
• Ex) logbx3 logby

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See Example 1, pg. 401

• Express as a single logarithm

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Check Point 1

• Use the product rule to expand each logarithmic
  expression
• A) log6(7 11) B) log(100x)

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QUOTIENT RULE, page 401

• Quotient Property
• logb(M/N) logbM logbN
• The logarithm of a quotient is the logarithm of
  the numerator minus the logarithm of the
  denominator.
• Ex) log2w - log216

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See Example 2, page 402.

• Express as a difference of logarithms.

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Check Point 2

• Use the quotient rule to expand each logarithmic
  expression

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POWER RULE, page 402

• Power Property logbMp p logbM
• The logarithm of a power of M is the exponent
  times the logarithm of M.
• Ex) log2x3

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See Example 3, page 403.

• Express as a product.

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Check Point 3

• Use the power rule to expand each logarithmic
  expression

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Extra Practice

• Express as a product.

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Expanding Logarithmic Expressions(See blue box
on page 403.)

• Use properties of logarithms to change one
  logarithm into a sum or difference of others.
• Example

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See Example 4, page 404

• Check Point 4 Use log properties to expand each
  expression as much as possible.

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Expanding Logs Express as a sum or difference.
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More Practice Expanding

• log27b
• log(y/3)2

• c) log7a3b4

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Condensing Logarithmic Expressions(See blue box
on page 404.)

• We can also use the properties of logarithms to
  condense expressions or write as a single
  logarithm.
• See Example 5, page 404.

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Lets reverse things.

• Express as a single logarithm.

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Pencils down. Watch and listen.

• Express as a single logarithm.
• Solution

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Check Point 5

• Write as a single logarithm.

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Check Point 6Write as a single logarithm.
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Check Point 6Write as a single logarithm.
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More Practice

• d) Write 3log2 log 4 log 16 as a single
  logarithm.
• e) Can you write 3log29 log69 as a single
  logarithm?

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Review of Properties(from Lesson 3.2)

• The Logarithm of a Base to a Power
• For any base a and any real number x,
• loga a x x.
• (The logarithm, base a, of a to a power is the
  power.)
• A Base to a Logarithmic Power
• For any base a and any positive real number x,
• (The number a raised to the power loga x is x.)

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Examples

• Simplify.
• a) loga a 6
• b) ln e ?8

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Simplify.

• A)
• B)

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Change of Base Formula

• The 2 bases we are most able to calculate
  logarithms for are base 10 and base e. These are
  the only bases that our calculators have buttons
  for.
• For ease of computing a logarithm, we may want to
  switch from one base to another using the formula

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See Examples 7 8, page 406-7.

• Check Point 7 Use common logs to evaluate log7
  2506.
• Check Point 8 Use natural logs to evaluate log7
  2506.

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Summary of Properties of Logarithms
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Summary of Properties of Logarithms (cont.)