Properties

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Properties Laws of Logarithms
Learning Goals

• Use properties and laws of logarithms to simplify
  and evaluate expressions.

2
Important Idea
The definitions of common and natural logarithms
differ only in their bases, therefore, they share
the same properties and laws.
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Important Idea
Properties of Common Logarithms

• log v defined only for vgt0

• log 10 log 101

4
Important Idea
Properties of Natural Logarithms

• ln v defined only for vgt0

• ln 10 ln e1

• for vgt0

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Example
Use the properties of logarithms to solve the
equation
6
Try this
Use the properties of logarithms to solve the
equation
x13
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Important Idea
Product Law
ln(ab) ln a ln b
Quotient Law
ln ann ln a
Power Law
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Example
Find given
and
What law was used?
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Example
Find given
and
What law was used?
10
Try This
Using the product law,
find given
and
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Example
Using the product law, write the given expression
as a single logarithm
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Try This
Using the product law, write the given expression
as a single logarithm
13
Try This
Using the product law, write the given expression
as two logarithms
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Example
Find given
and
What law was used?
15
Try This
Find given
and
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Example
Using the quotient law, write the given
expression as a single logarithm
17
Try This
Using the quotient law, write the given
expression as a single logarithm
18
Try This
Using the quotient law, write the given
expression as two logarithms
19
Example
Using the power law, re-write the given
expression and simplify if possible
20
Example
Using the power law, re-write the given
expression and simplify if possible
21
Try This
Using the power law, re-write the given
expression and simplify if possible
22
Example
Use a combination of logarithmic properties and
laws to re-write the given expression
23
Example
Use a combination of logarithmic properties and
laws to re-write the given expression
24
Try This
Use a combination of logarithmic properties and
laws to re-write the given expression
25
Example
The 1989 world series earthquake in San Francisco
measure 7.0 on the Richter Scale. The great
earthquake of 1906 measured 8.3. How much more
intense was the 1906 quake?
26
Example
Decibels are calculated by the function
where is the minimum sound intensity
detectable by the human ear. Find the decibel
level of a jet engine which is 10 billion times
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Lesson Close
The manipulation of logarithms is a fundamental
math skill that you will need in upper level math
courses and in science and engineering.