LOGARITHMIC FUNCTIONSPresented by:

AMEENA AMEEN MARYAM BAQIRFATIMA EL MANNAIKHOLOODREEM IBRAHIMMARIAM 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

• Binary logarithm• Logarithmic Equation• The natural logarithm• Proof that d/dx ln(x) =1/x• Graphing logarithmic functions.

AMEENA

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 exponential equation could be written in terms of a logarithmic equation as this form

• a^ y = Х Loga x = y

Example of logarithms:

416log2

1

5243log3

216

1log4

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:-

3^3=27

2^-5=1/32

4^0=1

MARYAM.B

Properties of Logarithm

• because

• because

• becausexa xa log

1log aa

01log a 10 a

aa 1

xaxa

Property1: loga1=0 because a0=1

• Examples:

• (a) 90=1

• (b) log91=0

Property 2: logaa=1 because a1=a

• Examples:

• (a) 21=2

• (b)log22=1

Property 3:logaax=x because ax=ax

• Examples:

• (a) 24=24

• (b) log224=4

• (c) 32=9 log39=2log332=2

Property4:blogbx=x

• Example:

• 3log35=5

FATIMA

There are three laws of logarithms:

nmmn aaa logloglog

nmn

maaa logloglog

mnm an

a loglog

1

2

3

Logarithm of products

Logarithm of quotient

Logarithm of a power

Remember these laws:

01log a

1log aa

1

2

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

nmmn aaa logloglog

Example:

3log5log aa

)35(log a

15loga

Transform the addition into multiplication

nmn

maaa logloglog

Example2:7log35log aa

)7

35(loga

5loga

Transforming the subtraction into division

mnm an

a loglog

Example3:

8log24log3 aa

4096log

64log2

64log64log

a

a

aa

4log3 a

3)4(loga

8log2 a

2)8(loga

The form of

Will be changed into

And the same for

Will be

Solve it yourself!

3log24log5log2 aaa

22 3log4log5log aaa

9

111log

9

425log

9

425log

a

a

a

KHOLOOD

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

Changing the base:

xba log

ba x

ba cx

c loglog

bax cc loglog

a

bx

c

c

log

log

let

Divide each side by

Take the base-c logarithm of each side

Power rule

aclog

a

bb

c

ca log

loglog

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

If the new base is 10 or e, then:*

)ln(

)ln(

)log(

)log()(log

a

b

a

bba

Common logarithm:

In mathematics,the common logarithm is the logarithm with base 10. It is also

known as the decadic logarithm, [] .

Examples:

Binary logarithm:

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

Examples:

n2

n2log

Binary logarithm:

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

Examples:

n2n2log

REEM

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 LogarithmThe 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 LogarithmThe natural logarithm function can

also be defined as the inverse function of the exponential function, leading to the identities:

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.

For Example

Property of Logarithms:

Definition of Logarithm

8 = x- 7x Simplify

0 = x- 7x – 8 Write quadratic equation in standard form

0 = (x – 8)(x + 1) Solve by factoring

x – 8 = 0 or x + 1 = 0x = 8 or x = -1

Substitute the solution

8 for x

Substitute the solution –1 for x

Subtract Subtract

3 + 0 = 3 Because ;

because

The number -1 does not check, since negative numbers do not have logarithm

s 3=3

Proof that d/dx ln(x) = 1/xThe 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

MARIAM QAROOT

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:

• 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..

01log by(b,1)

b1

1

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:

10 xFractions are the values of the negative powers.

Examples on graphing logarithm:• EXAMPLE ONE

Graph y = log2(x).First change log to exponent

form:X=2 power y, then start with

a T-chart

X Y= LOG 2(X)

0.125 log2(0.125) = –3

0.5 log2(0.5) = –1

1 log2(1) = 0

2 log2(2) = 1

4 log2(4) = 2

8 log2(8) = 3

• EXAMPLE two:

GraphFirst change ln into logarithm

form:Loge (x)Then change to exponential

form:X= e power y..Now draw you T-

chart

xy ln

X Y= loge (x)

0.13 -2

0.36 -1

1 0

2.71 1

7.38 2

• EXAMPLE two: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 shiftingThe 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 leftdraw the asymptote on the x= -3

The graph of y = log2(x + 3) Looks like this:

Remember:

• You may get some question about log like for example:

• Log2 )x+15( = 2

• Solution:

• 2^2= x+15x= -11, which can never be realTherefore, No Solution

Compare between logarithm and exponential graphs:

xxf 3)(

xxf 31 log)(

The Equations  y = b x  and  x = log b y  say the same thing.

y = loga x

1 2 3 4-1-2-3-4

x1

2

3

4

-1

-2

-3

-4

y

1 2 3 4-1-2-3-4

x1

2

3

4

-1

-2

-3

-4

y

y =- loga x

y = log2 (-x)

.

1 2 3 4-1-2-3-4

x1

2

3

4

-1

-2

-3

-4

y

1 2 3 4-1-2-3-4

x1

2

3

4

-1

-2

-3

-4

y

Y=loga(x+2)

Y=loga(x+2)

1 2 3 4-1-2-3-4

x1

2

3

4

-1

-2

-3

-4

y

y = loga (x-2)

1 2 3 4-1-2-3-4

x1

2

3

4

-1

-2

-3

-4

y

y = logax +2

1 2 3 4-1-2-3-4

x1

2

3

4

-1

-2

-3

-4

y

y = loga x -2

1 2 3 4-1-2-3-4

x1

2

3

4

-1

-2

-3

-4

y

THE ENDTHE END