Post on 25-Dec-2015
Logarithmic FunctionsLogarithmic Functions
TS:Making Decisions After TS:Making Decisions After Reflection and ReviewReflection and Review
ObjectivesObjectives
To write exponential equations in To write exponential equations in logarithmic form.logarithmic form.
To use properties of logarithms to expand To use properties of logarithms to expand and condense logarithmic expressions.and condense logarithmic expressions.
It is asking: It is asking:
““What power would I take b to in order to get a?”What power would I take b to in order to get a?”
Logarithmic FunctionsLogarithmic Functions
Key to understanding logarithms:Key to understanding logarithms:
A A logarithmlogarithm is an is an exponentexponent!!
logB A CExponent
Base Argument
CB A
Logarithmic FunctionsLogarithmic Functions
32 8 2log 8 3
25 25 5log 25 2
10 7x 10log 7 x
34 64 4log 64 3
5 125x 5log 125 x
Exponential FormExponential Form Logarithmic FormLogarithmic Form
Logarithmic FunctionsLogarithmic Functions
Evaluate:Evaluate: 4log 2 n
4 2n
2 12 2n
2 12 2n
2 1n
12n
Properties of LogarithmsProperties of Logarithms
ln ln lnAB A B
ln ln lnA
A BB
ln lnBA B A
ln xe x
ln xe x
Properties of LogarithmsProperties of Logarithms
Expand:Expand:23
lnx
y
2ln3 lnx y2ln3 ln lnx y
ln3 2ln lnx y
Properties of LogarithmsProperties of Logarithms
Expand:Expand: 2ln 1x x
2ln ln 1x x
ln 2ln 1x x
ln does not distribute!
ln 1 ln ln1x x
Properties of LogarithmsProperties of Logarithms
Expand:Expand:2
3ln
6
x
y
2 3ln ln 6x y
2 3ln ln 6 lnx y
2ln ln 6 3lnx y
Properties of LogarithmsProperties of Logarithms
Expand:Expand: 2
3ln xy
1
22
3ln xy
2
312 ln x
y
2 312 ln lnx y
12 2ln 3lnx y
32ln lnx y
ConclusionConclusion
A logarithm indicates the exponent to which you A logarithm indicates the exponent to which you raise a certain base in order to produce a given raise a certain base in order to produce a given value.value.
The inverse of logarithmic function is an The inverse of logarithmic function is an exponential function.exponential function.
Logs to the base 10 are written without a base.Logs to the base 10 are written without a base.
Logs to the base Logs to the base ee are indicated by the symbol are indicated by the symbol lnln..
Begin your HW –Day 7 p.283 #1-8, 23-39Begin your HW –Day 7 p.283 #1-8, 23-39
Re-write the logarithmic Re-write the logarithmic equation as an exponential equation as an exponential equation, or vise versa.equation, or vise versa.
1)1)
2)2)
3)3)
4)4)
5)5)
6)6)
7)7)
8) 8)
ln8.4 2.1282
ln 0.056 2.8824
ln 2 0.6931
ln 0.2 1.6094
0 1e 2 7.3891e 3 0.0498e
0.25 1.2840e
Apply the inverse properties of Apply the inverse properties of logarithmic and exponential logarithmic and exponential functions to simplify.functions to simplify.
23)23)
24)24)
25)25)
26)26)
27)27)
28)28)
2 1ln xe
21 ln xe
2
ln xe
ln(5 2)xe
ln xe3ln8 xe
Logarithmic FunctionsLogarithmic FunctionsDay 2Day 2
TS:Making Decisions After TS:Making Decisions After Reflection and ReviewReflection and Review
ObjectivesObjectives
To use properties of logarithms to expand To use properties of logarithms to expand and condense logarithmic expressions.and condense logarithmic expressions.
To be able to solve logarithmic and To be able to solve logarithmic and exponential equationsexponential equations
Properties of LogarithmsProperties of Logarithms
ln ln lnAB A B
ln ln lnA
A BB
ln lnBA B A
ln xe x
ln xe x
Properties of LogarithmsProperties of Logarithms
Combine:Combine: ln 1 ln 2 3lnx x x
3ln 1 2 lnx x x
2 3ln 3 2 lnx x x
2
33 2ln x xx
Properties of LogarithmsProperties of Logarithms
Combine:Combine: 4ln3 2ln lnx y
4 2ln3 ln lnx y
281ln lnx
y
281lnx y
Properties of LogarithmsProperties of Logarithms
Combine:Combine: 2122ln3 ln 1x
1
22 2ln3 ln 1x
2
9
1ln
x
Logarithmic FunctionsLogarithmic Functions
Solve:Solve: Solve:Solve:24 64x 2 34 4x 2 3x
2 7x
ln 2 ln 7x
ln 2 ln 7x 1x ln 7
ln 2x
2.807x
Logarithmic FunctionsLogarithmic Functions
Solve:Solve: Solve:Solve:34 9x 3ln 4 ln9x
( 3)ln 4 ln9x
2 10xe
5xe
ln ln5xe ln9ln 43x
ln9ln 4 3x
4.585x
ln5x
1.609x
Logarithmic FunctionsLogarithmic Functions
Solve:Solve: Solve:Solve:2 15 2 115xe 2 12 110xe 2 1 55xe
32 1.5 640x
1.5 20x
ln 1.5 ln 20x
2 1ln ln55xe 2 1 ln55x
ln 1.5 ln 20x ln 20ln1.5x
2 ln55 1x
1.504x
7.39xln55 12x
Logarithmic FunctionsLogarithmic Functions
Solve:Solve: Solve:Solve: 250 3 125xe 23 2.5xe
2 0.5xe
8ln 3 2 1.5x ln(3 2) 0.1875x
0.1875 3 2e x 2 0.5xe 2ln ln 0.5xe 2 ln 0.5x
0.1875 2 3e x 0.1875 2
3ex
ln 0.52x
0.35x
1.07x
Logarithmic FunctionsLogarithmic Functions
Suppose you deposit money into an account whose Suppose you deposit money into an account whose annual interest rate is 4% compounded continuously. annual interest rate is 4% compounded continuously. How long will it take for the money to double?How long will it take for the money to double?
rtA Pe0.042 tP Pe
0.042 te0.04ln 2 ln te
ln 2 0.04t17.3 yearst
ConclusionConclusion
A logarithm indicates the exponent to which you A logarithm indicates the exponent to which you raise a certain base in order to produce a given raise a certain base in order to produce a given value.value.
The inverse of logarithmic function is an The inverse of logarithmic function is an exponential function.exponential function.
Logs to the base 10 are written without a base.Logs to the base 10 are written without a base.
Logs to the base Logs to the base ee are indicated by the symbol are indicated by the symbol lnln..
Begin your HW –Day 8 p.284 #41-63,67, Begin your HW –Day 8 p.284 #41-63,67, 71-77odd71-77oddWrite as a single logarithm.Write as a single logarithm.
41)41)
42)42)
43)43)
44)44)
45)45)
46)46)
47)47)
48)48)
49)49)
50) 50)
ln(2 1) ln(2 1)x x
21 [2ln( 3) ln ln( 1)]3 x x x
ln( 2) ln( 2)x x
3ln 2ln 4lnx y z
3[ln ln( 3) ln( 4)]x x x 212ln3 ln( 1)2 x
23 [ln ( 1) ln( 1)]2 x x x 12ln ln( 1)2x x
Solve for x or t.Solve for x or t.
51)51)
52)52)
53)53)
54)54)
55)55)
56)56)
57)57)
58)58)
59)59)
60)60)
2ln 9 0xe
2ln 4x
ln 4xe
ln 0x
1 4xe 0.5 0.075xe
0.2300 700te
2[ln ln( 1)] 3[ln ln( 1)]x x x x 31 ln( 2) ln( 2)2 2x x
0.0174 0.5te 25 15x
400(1.06) 1300t