4'3 Properties of Logarithms

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4.3 Properties of Logarithms
MAC 1140 Mrs. Kessler
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The Product Rule

• Let b, M, and N be positive real numbers with b
  ? 1.
• logb (MN) logb M logb N

In other words, The logarithm of a product is the
sum of the logarithms.
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The Quotient Rule

• Let b, M and N be positive real numbers with b ?
  1.
• In other words,
• The logarithm of a quotient is the difference of
  the logarithms.

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The Power Rule

• Let b, M, and N be positive real numbers with b
  1, and let p be any real number.
• In other words,
• The logarithm of a number with an exponent is the
  product of the exponent and the logarithm of that
  number.

log b M p p log b M
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Example 1
Write as a single logarithm log4 2 log4 32
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Example 2
Write as a single logarithm 1. ln x ln y
2. 2ln x ln y
3. ½( ln x ln y)
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Summary of Properties for Expanding Logarithmic
Expressions

• For M gt 0 and N gt 0

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Example 3a

• Use logarithmic properties to expand the
  expression as much as possible.

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Example 3b

• Use logarithmic properties to expand the
  expression as much as possible.

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Summary ofProperties for Condensing Logarithmic
Expressions

• For M gt 0 and N gt 0

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Condense the Logarithmic Expressions
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The Change-of-Base Property

• For any logarithmic bases a and b, and any
  positive number M,
• The logarithm of M with base b is equal to the
  logarithm of M with any new base divided by the
  logarithm of b with that new base.

Since calculators only work with base 10 and base
e, this formula becomes very helpful.
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Example 4
The Change-of-Base Property

• Use logarithms to evaluate log37.

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Example 5
The Change-of-Base Property
y log x in red

• Graph y log3x.

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Example 5
The Change-of-Base Property

• Graph y log3x.
• How do you do this on a graphing calculator?

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Example 6

• Graph y log3x by hand. We have already y
  using a table. Is there another way?

What is the inverse function to y log3x ?
y 3x This is easy to graph
Tadah!!!
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Please read text and do HW.