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Logarithmic Functions
TS:Making Decisions After
Reflection and Review
Objectives
To write exponential equations in
logarithmic form.
To use properties of logarithms to expand
and condense logarithmic expressions.
Logarithmic Functions
Key to understanding logarithms:
A logarithm is an exponent!
log B A  C
Base
Exponent
Argument
BC  A
It is asking:
“What power would I take b to in order to get a?”
Logarithmic Functions
Exponential Form
Logarithmic Form
2 8
log 2 8  3
52  25
log5 25  2
10 x  7
log10 7  x
43  64
log 4 64  3
5  125
log5 125  x
3
x
Logarithmic Functions
Evaluate:
log3 9  n
3n  9
n2
Logarithmic Functions
Evaluate:
log5 1  n
5n  1
n0
Logarithmic Functions
Evaluate:
log 4 2  n
4 2
n
2 
2 n
2
1
2 2
2n
1
2n  1
n  12
Logarithmic Functions
Evaluate:
log5 5  n
5n  5
n 1
Logarithmic Functions
Evaluate:
log 5 53  n
5n  53
n3
Logarithmic Functions
Evaluate:
log10 1000  n
10n  1000
n3
Logarithmic Functions
Evaluate:
log 0.01  n
1
10  100
n
10  1012
n
10  10
n
n  2
2
Special Bases
log10 A  log A
Common log
log e A  ln A
Natural log
Natural Logarithm
Evaluate:
ln1  0
ln e  1
ln e 4  4
Properties of Logarithms
ln AB  ln A  ln B
A
ln  ln A  ln B
B
ln A  B ln A
B
ln e  x
x
e
ln x
x
Properties of Logarithms
Expand:
3x 2
ln
y
ln 3x 2  ln y
ln 3  ln x 2  ln y
ln 3  2ln x  ln y
Properties of Logarithms
Expand:
ln x  x  1
2
ln x  ln  x  1
2
ln x  2ln  x  1
ln does not distribute!
ln  x  1  ln x  ln1
Properties of Logarithms
Expand:
 x2 
ln  3 
 6y 
ln x 2  ln 6 y 3
ln x 2   ln 6  ln y 3 
2ln x  ln 6  3ln y
Properties of Logarithms
Expand:
x2
y3
ln
ln
1
2
1
2
1
2
 
2
x
y3
ln
1
2
x2
y3
2
3
ln
x

ln
y


 2ln x  3ln y 
ln x  32 ln y
Conclusion
A logarithm indicates the exponent to which you
raise a certain base in order to produce a given
value.
The inverse of logarithmic function is an
exponential function.
Logs to the base 10 are written without a base.
Logs to the base e are indicated by the symbol ln.
Begin your HW –Day 7 p.283 #1-8, 23-39
Re-write the logarithmic
Apply the inverse properties of
equation as an exponential
logarithmic and exponential
equation, or vise versa.
functions to simplify.
1) ln 2  0.6931
2) ln8.4  2.1282
3) ln 0.2  1.6094
4) ln 0.056  2.8824
5) e 0  1
6) e 2  7.3891
7) e 3  0.0498
8) e0.25  1.2840
23)
24)
25)
26)
27)
28)
x2
ln e
2 x 1
ln e
eln(5 x2)
1  ln e2 x
ln x
e
ln x3
8  e
Logarithmic Functions
Day 2
TS:Making Decisions After
Reflection and Review
Objectives
To use properties of logarithms to expand
and condense logarithmic expressions.
To be able to solve logarithmic and
exponential equations
Properties of Logarithms
ln AB  ln A  ln B
A
ln  ln A  ln B
B
ln A  B ln A
B
ln e  x
x
e
ln x
x
Properties of Logarithms
Combine:
ln 4  ln x
ln 4x
Properties of Logarithms
Combine:
2ln 8  5ln z
ln82  ln z 5
ln 64z 5
Properties of Logarithms
Combine: ln  x  1  ln  x  2   3ln x
ln  x  1 x  2   ln x
ln  x  3 x  2   ln x
2
ln

x 2 3 x  2
x3

3
3
Properties of Logarithms
Combine:
4ln 3  2ln x  ln y
ln 3  ln x  ln y
4
2
ln 81
2  ln y
x
ln x812 y
Properties of Logarithms
Combine:
2ln 3  ln  x  1
2
1
2
ln 3  ln  x  1
2
2
ln
9
x 2 1
1
2
Logarithmic Functions
Solve: 4
x 2
 64
4 x2  43
x23
x 1
Solve:
2x  7
ln 2  ln 7
x
x ln 2  ln 7
x  lnln 72
x  2.807
Logarithmic Functions
Solve:
4
x3
9
ln 4 x3  ln 9
( x  3)ln 4  ln 9
x  3  lnln94
x  lnln 94  3
x  4.585
Solve:
2e x  10
e 5
x
ln e x  ln 5
x  ln 5
x  1.609
Logarithmic Functions
Solve: 5  2e
2e
2 x 1
2 x 1
 115
 110
e 2 x1  55
2 x 1
ln e
 ln 55
2 x  1  ln 55
2 x  ln 55  1
x  ln 552 1
x  1.504
Solve: 32 1.5  640
x
1.5  20
x
ln 1.5  ln 20
x ln 1.5  ln 20
x
ln 20
x  ln1.5
x  7.39
Logarithmic Functions


Solve: 50 3  e 2 x  125
3  e  2.5
e 2 x  0.5
2x
e  0.5
2x
ln e  ln 0.5
2 x  ln 0.5
x  ln20.5
x  0.35
2x
Solve: 8ln  3x  2   1.5
ln(3x  2)  0.1875
e0.1875  3 x  2
e
0.1875
x
 2  3x
e0.1875  2
3
x  1.07
Logarithmic Functions
Suppose you deposit money into an account whose
annual interest rate is 4% compounded continuously.
How long will it take for the money to double?
A  Pert
2 P  Pe
0.04 t
2  e0.04t
ln 2  ln e0.04t
ln 2  0.04t
t  17.3 years
Conclusion
A logarithm indicates the exponent to which you
raise a certain base in order to produce a given
value.
The inverse of logarithmic function is an
exponential function.
Logs to the base 10 are written without a base.
Logs to the base e are indicated by the symbol ln.
Begin your HW –Day 8 p.284 #41-63,67,
71-77odd
Write as a single logarithm.
Solve for x or t.
41) ln( x  2)  ln( x  2)
42) ln(2 x  1)  ln(2 x  1)
43) 3ln x  2ln y  4ln z
2
1
[2ln(
x

3)

ln
x

ln(
x
 1)]
44) 3
45) 3[ln x  ln( x  3)  ln( x  4)]
46) 2ln 3  1 2 ln( x 2  1)
47) 3 2 [ln x( x 2  1)  ln( x  1)]
48) 2ln x  1 2 ln( x  1)
49) 2[ln x  ln( x  1)]  3[ln x  ln( x  1)]
50) 1 2 ln( x  2)  3 2 ln( x  2)
51) eln x  4
2
ln
x
52) e
9  0
53) ln x  0
54) 2ln x  4
55) e x1  4
56) e 0.5 x  0.075
57) 300e 0.2t  700
58) e 0.0174t  0.5
59) 52 x  15
60) 400(1.06)t  1300