1. Warm-Up 4/22 A = 3 β 5 2
Dec 13, 2015
Rigor:You will learn how to evaluate, graph and use the
properties of logarithmic functions.
Relevance:You will be able to solve pH chemistry problems
using logarithmic functions.
Example 1: Evaluate each logarithm.
a. b. log 3 81=π₯
3π₯ΒΏ 813π₯ΒΏ 34
π₯ΒΏ 4
log 5 β5=π₯5π₯ΒΏ β55π₯ΒΏ5
12
π₯ΒΏ12
log 3 81=4 log 5 β5=12
c. d. log 7
149
=π₯
7π₯ΒΏ149
7π₯ΒΏ7β2
log 7149
=β2
log 22=π₯2π₯ΒΏ2
log 22=1
Example 3: Evaluate each expression.
a. b. ΒΏ log10β3
ΒΏβ3ΒΏ5
c. d. β1.4149 undefinedU se acalculator .
Example 5a: Sketch and analyze the graph of the function. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing, or decreasing.
π₯=log 3 π¦3π₯=π¦
x
β 3 .037
β 2 .111
β 1 .333
0 1
1 3
2 9
3 27
π (π₯ )=log3 π₯x y
x
.037 β 3
.111 β 2
.333 β 1
1 0
3 1
9 2
27 3
π β1 (π₯ )=3π₯
π (π₯ )=log3 π₯
Domain:Range:x-intercept:Asymptotes:End Behavior:
Increasing/Decreasing:
(0 ,β)(ββ ,β)
(1 ,0)π₯=0
limπ₯β0+ΒΏ π ( π₯)=ββΒΏ
ΒΏ limπ₯ββ
π (π₯)=βand
I ncreasing :(0 ,β)
Example 5b: Sketch and analyze the graph of the function. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing, or decreasing.
π₯=log 12
π¦12
π₯
=π¦
x
β 3 8
β 2 4
β 1 2
0 1
1 .5
2 .25
3 .125
π (π₯ )= log 12
π₯
x y
x
8 β 3
4 β 2
2 β 1
1 0
.5 1
.25 2
.125 3
πβ1 (π₯ )=12
π₯
π (π₯ )=log 12
π₯
Domain:Range:x-intercept:Asymptotes:End Behavior:
Increasing/Decreasing:
(0 ,β)(ββ ,β)
(1 ,0)π₯=0
limπ₯β0+ΒΏ π ( π₯)=β ΒΏ
ΒΏ limπ₯ββ
π (π₯)=ββand
Decreasing :(0 ,β)
Example 6a: Use the graph of to describe the transformation of the function. Then sketch both functions.
π (π₯ )=log (π₯+4 )
x y
β 4 V.A.
β 3 0
β 2 .30103
β 1 .47712
0 .60206
1 .69897
2 .77815
π (π₯ )=log π₯
x y
0 V.A.
1 0
2 .30103
3 .47712
4 .60206
5 .69897
6 .77815
π (π₯ )= π (π₯+4)The graph of is the graph of translated 4 units to the left.
Example 6b: Use the graph of to describe the transformation of the function. Then sketch both functions.
π (π₯ )=β logπ₯β5
x y
0 V.A.
1 β 5
2 β 5.3010
3 β 5.4771
4 β 5.6020
5 β 5.6989
6 β 5.7781
π (π₯ )=log π₯
x y
0 V.A.
1 0
2 .30103
3 .47712
4 .60206
5 .69897
6 .77815
π (π₯ )=β π (π₯ )β5The graph of is the graph of is reflected in the x-axis and translated 5 units down.
Example 6c: Use the graph of to describe the transformation of the function. Then sketch both functions.
π (π₯ )=3 log (π₯+2 )
x y
β 2 V.A.
β 1 0
0 .90309
1 1.4314
2 1.8062
3 2.0969
4 2.3345
π (π₯ )=log π₯
x y
0 V.A.
1 0
2 .30103
3 .47712
4 .60206
5 .69897
6 .77815
π (π₯ )=3 π (π₯+2)The graph of is the graph of is expanded vertically by a factor of 3 andtranslated 2 units to the left.