Logarithmic functions
This presentation is the property of its rightful owner.
Sponsored Links
1 / 61

Logarithmic Functions PowerPoint PPT Presentation


  • 184 Views
  • Uploaded on
  • Presentation posted in: General

PreCalculus NYOS Charter School Quarter 4 “If we did all the things we were capable of doing, we would literally astound ourselves .” ~ Thomas Edison. Logarithmic Functions. Logarithmic Functions. The logarithmic function y = log a x, where a > 0 and a ≠ 1,

Download Presentation

Logarithmic Functions

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Logarithmic functions

PreCalculusNYOS Charter SchoolQuarter 4“If we did all the things we were capable of doing, we would literally astound ourselves.” ~ Thomas Edison

Logarithmic Functions


Logarithmic functions1

Logarithmic Functions

  • The logarithmic function y = loga x,

    where a > 0 and a ≠ 1,

    is the inverseof the exponential function y = ax.

    y = loga x iff x = ay


Logarithmic functions2

Logarithmic Functions

Example: Write in exponential form.

log3 9 = 2


Logarithmic functions3

Logarithmic Functions

Example: Write in exponential form.

log3 9 = 2


Logarithmic functions4

Logarithmic Functions

Example: Write in exponential form.

log8 2 =


Logarithmic functions5

Logarithmic Functions

Example: Write in exponential form.

log8 2 =


Logarithmic functions6

Logarithmic Functions

Example: Write in exponential form.

log125 25 =


Logarithmic functions7

Logarithmic Functions

Example: Write in exponential form.

log125 25 =


Logarithmic functions8

Logarithmic Functions

Example: Write in logarithmic form.


Logarithmic functions9

Logarithmic Functions

Example: Write in logarithmic form.

log4 64 =


Logarithmic functions10

Logarithmic Functions

Example: Write in logarithmic form.


Logarithmic functions11

Logarithmic Functions

Example: Write in logarithmic form.

log3=


Logarithmic functions12

Logarithmic Functions

Example: Evaluate log7.

y = log7

y = -2


Logarithmic functions13

Logarithmic Functions

Example: Evaluate log5.

y = log5


Logarithmic functions14

Logarithmic Functions

Example: Evaluate log5.

y = log5

y = -3


Logarithmic functions15

Logarithmic Functions

Properties of Logarithms


Logarithmic functions16

Logarithmic Functions

Example: Expand log5 9x

= log5 9 + log5 x


Logarithmic functions17

Logarithmic Functions

Example: Expand logx12y


Logarithmic functions18

Logarithmic Functions

Example: Expand logx12y

= logx12 + logxy


Logarithmic functions19

Logarithmic Functions

Properties of Logarithms


Logarithmic functions20

Logarithmic Functions

Example: Expand log5 9/x

= log5 9 - log5 x


Logarithmic functions21

Logarithmic Functions

Example: Expand logx12/y


Logarithmic functions22

Logarithmic Functions

Example: Expand logx12/y

= logx12 - logxy


Logarithmic functions23

Logarithmic Functions

Properties of Logarithms


Logarithmic functions24

Logarithmic Functions

Example: Simplify log5 9x

= x log5 9


Logarithmic functions25

Logarithmic Functions

Properties of Logarithms


Logarithmic functions26

Logarithmic Functions

Example: Simplify. log5 9 = log5 x

9 = x


Logarithmic functions27

Logarithmic Functions

Example: Solve for x. log5 16 = log5 2x

16 = 2x

8 = x


Logarithmic functions28

Logarithmic Functions

Properties of Logarithms


Logarithmic functions29

Logarithmic Functions

Example: Simplify. log5 5

1


Logarithmic functions30

Logarithmic Functions

Example: Simplify. log87 87

1


Logarithmic functions31

Logarithmic Functions

Example: Simplify. log87 1

0


Logarithmic functions32

Logarithmic Functions

Example: Simplify. log48 1

0


Logarithmic functions33

Logarithmic Functions

Example: Solve. log8 48 – log8 w = log8 6


Logarithmic functions34

Logarithmic Functions

Example: Solve. log8 48 – log8 w = log8 6

log8(48/w) = log86

48/w = 6

w = 8


Logarithmic functions35

Logarithmic Functions

Example: Solve. log10= x


Logarithmic functions36

Logarithmic Functions

Example: Solve. log10= x

log10= x

x =


Logarithmic functions37

Logarithmic Functions

  • If a, b, and n are positive numbers and neither a nor b is 1, then the following is called the change of base formula:


Logarithmic functions38

Logarithmic Functions

Example: Rewrite with a base of 2.

log6 5

=


Logarithmic functions39

Logarithmic Functions

Example: Combine.

= log11 15


Logarithmic functions40

Logarithmic Functions

  • Natural logarithms have base e.

    ln 5


Logarithmic functions41

Logarithmic Functions

Example: Convert log6 254 to a natural logarithm and evaluate.

log6 254

=

≈ 3.09


Logarithmic functions42

Logarithmic Functions

Example: Convert log5 43 to a natural logarithm and evaluate.

log5 43


Logarithmic functions43

Logarithmic Functions

Example: Convert log5 43 to a natural logarithm and evaluate.

log5 43

=

≈ 2.34


Logarithmic functions44

Logarithmic Functions

Example: Solve using natural logs. 2x = 27

log2 27 = x

= x

x ≈ 4.75


Logarithmic functions45

Logarithmic Functions

Example: Solve. 9x-4 = 7.13


Logarithmic functions46

Logarithmic Functions

Example: Solve. 9x-4 = 7.13

log9 7.13 = x - 4

+ 4 = x

≈ 4.89


Logarithmic functions47

Logarithmic Functions

Example: Solve. 6x+2 = 14

The variable is in the exponent. Take the log of both sides.

ln6x+2 = ln 14


Logarithmic functions48

Logarithmic Functions

Example: Solve. 6x+2 = 14

ln6x+2 = ln 14

(x + 2) ln 6 = ln 14

x + 2 =

x ≈ -.53


Logarithmic functions49

Logarithmic Functions

Example: Solve. 2x-5 = 11


Logarithmic functions50

Logarithmic Functions

Example: Solve. 2x-5 = 11

ln2x-5= ln 11

(x – 5) ln 2 = ln11

x – 5 =

X ≈ 8.46


Logarithmic functions51

Logarithmic Functions

Example: Solve. 6x+2 = 14x-3


Logarithmic functions52

Logarithmic Functions

Example: Solve. 6x+2 = 14x-3

ln6x+2= ln14x-3

Move the exponents to the front and distribute…

x ln 6 + 2 ln6 = x ln 14 – 3 ln 14

Get the x terms on the left side and constants on the right…

x ln 6 - x ln 14 = – 3 ln14 – 2 ln 6

Factor out an x from the left side…

x (ln6 - ln 14) = – 3 ln 14 – 2 ln 6


Logarithmic functions53

Logarithmic Functions

Example: Solve. 6x+2 = 14x-3

x (ln 6 - ln 14) = – 3 ln 14 – 2 ln 6

x ≈ 13.57


Logarithmic functions54

Logarithmic Functions

  • Sometimes we may want to know how long it takes for a quantity modeled by an exponential function to double.


Logarithmic functions55

Logarithmic Functions

Why ?

N = N0ekt


Logarithmic functions56

Logarithmic Functions

Why ?

N = N0ekt

2N0= N0ekt

2 = ekt

ln 2 = lnekt

ln 2 = kt


Logarithmic functions57

Logarithmic Functions

Example: As a freshman in college, McKayla received $4,000 from her great aunt. She invested the money and would like to buy a car that costs twice that amount when she graduates in four years. If the money is invested in an account that pays 9.5% compounded continuously, will she have enough money for the car?


Logarithmic functions58

Logarithmic Functions

Example: $4,000; 9.5%; double in 4 yrs?


Logarithmic functions59

Logarithmic Functions

Example: What interest rate is required for an amount to double in 4 years?


Logarithmic functions60

Logarithmic Functions

Example: What interest rate is required for an amount to double in 4 years?

k ≈ 17.33%


  • Login