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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.

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### LOGARITHMIC FUNCTIONSPresented by:AMEENA AMEENMARYAM 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.
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
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

Properties of Logarithm

• because
• because
• because

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=2log332=2

Property4:blogbx=x

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

Logarithm of products

1

Logarithm of quotient

2

Logarithm of a power

3

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

Example:

into multiplication

Example2:

Transforming the subtraction

into division

The form of

Will be changed into

And the same for

Will be

Example3:

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

*

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:

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:

Binary logarithm:

In mathematics,

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

Examples:

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 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 Logarithm

The 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.

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!!
• 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..

(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:
• EXAMPLEONE

Graph y = log2(x).

First change log to exponent form:

EXAMPLEtwo:

Graph

First change ln into logarithm form:

Loge (x)

Then change to exponential form:

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

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:
• 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

y =- loga x

y = loga x

Y=loga(x+2)

y = log2 (-x)

.

y = loga (x-2)

Y=loga(x+2)

y = loga x -2

y = logax +2