3.18
SECTION 3.3: (MORE) PROPERTIES OF LOGS
PART A: READING LOG EXPRESSIONS
We will use grouping symbols as a means of clarifying the order of operations in
expressions.
Often, grouping symbols are omitted when they could have helped.
How do we read log expressions in those cases?
We use ln for convenience, but any log function with any nice base is dealt with
similarly.
Example: Read
ln3
x
+
7
as:
ln 3
x
( )
+
7
Example: Read
ln3
x
4
as:
ln 3
x
4
( )
Example: Read
ln
x
4
as:
ln
x
4
( )
,
not
as
ln
x
( )
4
Generally speaking, absent any grouping symbols, if we see “ln” or “log” followed by a
product, quotient, power, or mixture of the above, the “ln” applies to the whole
expression that follows. However, + and
−
signs that introduce new terms tend to
terminate the ln expression.
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PART B: LOG PROPERTIES BASED ON LAWS OF EXPONENTS
Remember, logs are exponents. The laws for exponents imply laws for logs.
We will state these properties using ln, though they apply to any log function with a nice
base.
If we use the rules from lefttoright, we are “expanding” the expression.
If we use the rules from righttoleft, we are “condensing” the expression.
Assume
A
>
0
and
B
>
0
.
A
and
B
may represent constants or variable expressions.
Product Rule
ln
AB
( )
=
ln
A
+
ln
B
Think: The log of a
product
equals the
sum
of the logs.
(We go one step down in the order of operations if we read lefttoright.)
Related Exponent Law: When multiplying powers of
e
, the exponent on the
product equals the sum of the exponents (of the factors):
e
A
e
B
=
e
A
+
B
Quotient Rule
ln
A
B
⎛
⎝
⎜
⎞
⎠
⎟
=
ln
A
−
ln
B
Think: The log of a
quotient
equals the
difference
of the logs.
Related Exponent Law: When dividing powers of
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 Fall '11
 staff
 Calculus, Order Of Operations, Derivative, Logarithm

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