Logarithmic Functions TS:Making Decisions After Reflection and Review.

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?”


Key to understanding logarithms:Key to understanding logarithms:

A A logarithmlogarithm is an is an exponentexponent!!

logB A CExponent

Base Argument

CB A


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


Evaluate:Evaluate: 3log 9 n

3 9n

2n



5 1n

0n



4 2n

2 12 2n

2 12 2n

2 1n

12n



5 5n

1n



35 5n

3n



10 1000n

3n


Evaluate:Evaluate: log0.01 n

110010n

21

1010n

210 10n

2n

Special BasesSpecial Bases

10log logA A

log lne A A

Common log

Natural log

Natural LogarithmNatural Logarithm

Evaluate:Evaluate: ln1

lne

4ln e

0

1

4

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


Expand:Expand:23

lnx

y

2ln3 lnx y2ln3 ln lnx y

ln3 2ln lnx y


Expand:Expand: 2ln 1x x

2ln ln 1x x

ln 2ln 1x x

ln does not distribute!

ln 1 ln ln1x x


Expand:Expand:2

3ln

6

x

y

2 3ln ln 6x y

2 3ln ln 6 lnx y

2ln ln 6 3lnx y


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


ln ln lnAB A B

ln ln lnA

A BB

ln lnBA B A

ln xe x

ln xe x


Combine:Combine: ln 4 ln x

4ln x


Combine:Combine: 2ln8 5ln z

2 5ln8 ln z

5ln 64z


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


Combine:Combine: 4ln3 2ln lnx y

4 2ln3 ln lnx y

281ln lnx

y

281lnx y


Combine:Combine: 2122ln3 ln 1x

1

22 2ln3 ln 1x

2

9

1ln

x


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


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


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


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


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

Logarithmic Functions TS:Making Decisions After Reflection and Review.

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