logarithmic functions n.
Download
Skip this Video
Download Presentation
Logarithmic Functions

Loading in 2 Seconds...

play fullscreen
1 / 61

Logarithmic Functions - PowerPoint PPT Presentation


  • 259 Views
  • Uploaded on

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,

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Logarithmic Functions' - tab


Download Now 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%