slide1 n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Content: PowerPoint Presentation
Download Presentation
Content:

Loading in 2 Seconds...

play fullscreen
1 / 53

Content: - PowerPoint PPT Presentation


  • 156 Views
  • Uploaded on

LOGARITHMIC FUNCTIONS Presented by: AMEENA AMEEN MARYAM BAQIR FATIMA EL MANNAI KHOLOOD REEM IBRAHIM MARIAM OSAMA. Content:. Definition of logarithm How to write a Logarithmic form as an Exponantional form Properties of logarithm Laws of logarithm Changing the base of log Common logarithm.

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 'Content:' - talib


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
slide1

LOGARITHMIC FUNCTIONSPresented by:AMEENA AMEENMARYAM BAQIRFATIMA EL MANNAIKHOLOODREEM IBRAHIMMARIAM OSAMA

content
Content:
  • Definition of logarithm
  • How to write a Logarithmic form as an Exponantional form
  • Properties of logarithm
  • Laws of logarithm
  • Changing the base of log
  • Common logarithm
slide3
Binary logarithm
  • Logarithmic Equation
  • The natural logarithm
  • Proof that d/dx ln(x) =1/x
  • Graphing logarithmic functions.
definition of logarithmic function
Definition of Logarithmic Function

The power to which a base must be raised to yield a given number

e.g.

the logarithm to the base 3 of 9, or log3 9, is 2, because 32 = 9

the general form of logarithm
The general form of logarithm:
  • The exponential equation could be written in terms of a logarithmic equation as this form
  • a^y= Х Loga x = y
common logarithms
Common logarithms:
  • The two most common logarithms are called (common logarithms)
  • and( natural logarithms).Common logarithms have a base of 10
  • log x = log10x
  • , and natural logarithms have a base of e.
  • ln x =logex
exponential form
Exponential form:-

3^3=27

2^-5=1/32

4^0=1

slide11

Properties of Logarithm

  • because
  • because
  • because
slide12

Property1: loga1=0 because a0=1

  • Examples:
  • (a) 90=1
  • (b) log91=0
slide13

Property 2: logaa=1 because a1=a

  • Examples:
  • (a) 21=2
  • (b)log22=1
slide14

Property 3:logaax=x because ax=ax

  • Examples:
  • (a) 24=24
  • (b) log224=4
  • (c) 32=9  log39=2log332=2
slide15

Property4:blogbx=x

  • Example:
  • 3log35=5
there are three laws of logarithms
There are three laws of logarithms:

Logarithm of products

1

Logarithm of quotient

2

Logarithm of a power

3

remember these laws
Remember these laws:

The log of 1 is always equal to 0 but the log of a number which is similar to the base of log is always equal 1

1

2

slide19

Example:

Transform the addition

into multiplication

slide20

Example2:

Transforming the subtraction

into division

slide21

The form of

Will be changed into

And the same for

Will be

Example3:

slide24

Changing the base:

Let a, b, and x be positive real numbers such that and (remember x must be greater than 0). Then can be converted to the base b by the formula

let

Take the base-c logarithm of each side

Power rule

Divide each side by

slide25

*

If a and b are positive numbers not equal to 1 and M is positive,

then

slide26

*

If the new base is 10 or e, then:

slide27

Common logarithm:

In mathematics,

the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm,[ ] .

Examples:

slide28

Binary logarithm:

The binary logarithm is the logarithm for base 2. It is the inverse function of .

Examples:

slide29

Binary logarithm:

In mathematics,

the binary logarithm is the logarithm for base 2. It is the inverse function of .

Examples:

the nature of logarithm
The Nature of Logarithm

Is the logarithm to the base e, where e is an irrational constant approximately equal to 2.718281828459.

the nature of logarithm1
The Nature of Logarithm

The natural logarithm can be defined for all positive real numbers x as the area under the curve y = 1/t from 1 to x, and can also be defined for non-zero complex numbers.

the nature of logarithm2
The Nature of Logarithm

The natural logarithm function can also be defined as the inverse function of the exponential function, leading to the identities:

logarithm equation
Logarithm Equation

Logarithmic equations contain logarithmic expressions and constants. When one side

of the equation contains a single logarithm and the other side contains a constant, the

equation can be solved by rewriting the equation as an equivalent exponential equation using the definition of logarithm.

proof that d dx ln x 1 x
Proof thatd/dx ln(x) = 1/x

The natural log of x does not equal 1/x, however the derivative of ln(x) does:

The derivative of log(x) is given as:d/dx[ log-a(x) ] = 1 / (x * ln(a))where "log-a" is the logarithm of base a.

However, when a = e (natural exponent),

then log-a(x) becomes ln(x) and ln(e) = 1:

d/dx[ log-e(x) ] = 1 / (x * ln(e))

d/dx[ ln(x) ] = 1 / (x * ln(e))

d/dx[ ln(x) ] = 1 / (x * 1)

d/dx[ ln(x) ] = 1 / x

graphing logarithms is a piece of cake
Graphing logarithms is a piece of cake!!
  • Basics of graphing logarithm
  • Comparing between logarithm and exponential graphs
  • Special cases of graphing logarithm
  • The logarithm families.
graphing basics
Graphing Basics:
  • The important key about graphing in general, is to stick in your mind the bases for this graph.
  • For logarithm the origin of its graph is square-root graph..
slide41

(b,1)

1

1

b

Before graphing y= logb (x) we can start first with knowing the following:

The logarithm of 1 is zero (x=1), so the x-intercept is always 1, no matter what base of log was.

For example if we have:

b = 2 power 0 = 1

b = 3 power 0 = 1

b = 4 power 0 = 1

Values of x between 0 and 1 represent the graph below the x-axis when:

Fractions are the values of the negative powers.

examples on graphing logarithm
Examples on graphing logarithm:
  • EXAMPLEONE

Graph y = log2(x).

First change log to exponent form:

X=2 power y, then start with a T-chart

slide43
EXAMPLEtwo:

Graph

First change ln into logarithm form:

Loge (x)

Then change to exponential form:

X= e power y..Now draw you T-chart

slide44
EXAMPLEtwo:

Graph y = log2(x + 3).

This is similar to the graph of log2(x),

but is shifted

"+ 3" is not outside of the log,

the shift is not up or down

First plot (1,0), test the shifting

The log will be 0 when the argument,

x + 3, is equal to 1.

When x = –2. (1, 0) the basic point is shifted to

(–2, 0)

So, the graph is shifted three units to the left

draw the asymptote on the x= -3

remember
Remember:
  • You may get some question about log like for example:
  • Log2 (x+15) = 2
  • Solution:
  • 2^2= x+15
  • x= -11, which can never be real
  • Therefore, No Solution
slide49

y =- loga x

y = loga x

slide50

Y=loga(x+2)

y = log2 (-x)

.

slide51

y = loga (x-2)

Y=loga(x+2)

slide52

y = loga x -2

y = logax +2