21: Lecture Section 3.2
Logarithmic Functions
Recall the graph of exponential function ( ) =
> 1:
,
(0,1)
Since ( ) is one-to-one, it has an inverse function.
Def. The logarithmic function with base ,
where > 0 and = 1 is written ( ) = log and
is dened by the relationship
= log
if and only if
ex. Write in exponential form:
1) log3
1
9
= 2
2) log (3 + 1) = 2
ex. Write in logarithmic form: 41.5 = 8
ex. Evaluate:
1) log2 64 =
2) log3 1 =
3) log3 3 =
4) log10
1
1000
5) log16 4 =
6) log2(1) =
7) log2 0 =
=
Properties of Logarithms
1. Recall: If
That is,
= log
, then
=
> 0.
2. log 1 =
3. log
=
4. Inverse Properties:
log
log
=
=
for all real number
for
>0
5. One-to-One Properties:
If log
= log
, then
ex. Evaluate:
1) 2log2 3 =
2) log5
1
=
5
ex. Solve: log2( 3) = log2 9
Graphs of Logarithmic Functions
ex. Sketch
= 2 and
ex. Sketch
=
1
2
= log2 .
and
= log 1
2
Properties of the graph of ( ) = log
Compare ( ) = log
and
1
( ) = log
1. Domain:
2. Range:
3. Intercept:
4. Asymptote:
5. increasing if
decreasing if
6. points on
the graph
ex. Graph ( ) = log3( + 1)
( )=
The )=
:
1
( Natural Logarithmic Function
= log
= ln
if and only if
Note the following:
ln 1 =
ln =
Inverse Properties:
ln
=
ln( ) =
One-to-One Property:
If ln
= ln , then
ex. Evaluate:
1)
ln(2 +3)
2) ln
1
=
=
ex. Solve: ln(
2
) = ln 6
ex. Graph and nd the domain and vertical asymptote of ( ):
1) ( ) = ln
2) ( ) = ln( 2) + 1
3) ( ) = ln( ) + 2
Common Logarithm Function
= log10
= log
if and only if
ex. Evaluate:
log 1 =
log 10 =
log 10000 =
1
log =
10
Applications
ex. The loudness level of a sound,
given by
= 10 log
1012
, in decibels, is
,
where is the intensity of a sound in watts per square
meter.
1) Determine the decibel level of a sound with an
intensity of 102 watt per square meter.
2) Determine the decibel level of a sound with an
intensity of 1 watts per square meter.
3) The intensity of a sound in part (2) is 100 times
as great as the intensity in part (1). By how much is
the decible level increased?
Practice.
Determine the domain of each function.
1) ( ) = log(
2
2) ( ) = log
3
+3
2 )
3) ( ) = ln 2
Answer.
1) (, 0) (2, ) 2) (3, 3) 3) (, 2) (2, )