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

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 Log a x = y

Example of logarithms :

Common logarithms: TThe two most common logarithms are called (common logarithms) aand( natural logarithms).Common logarithms have a base of 10 llog x = log 10 x ,, and natural logarithms have a base of e. lln x =log e x

Exponential form:- 3^3=27 2^-5=1/32 4^0=1

MARYAM.B

Properties of Logarithm because

Property1: log a 1=0 because a 0 =1 Examples: (a) 9 0 =1 (b) log 9 1=0

Property 2: log a a=1 because a 1 =a Examples: (a) 2 1 =2 (b)log 2 2=1

Property 3:log a a x =x because a x =a x Examples: (a) 2 4 =2 4 (b) log =4 (c) 3 2 =9  log 3 9=2  log =2

Property4:b log b x =x Example: 3 log 3 5 =5

FATIMA

There are three laws of logarithms: Logarithm of products Logarithm of quotient Logarithm of a power

Remember these laws: 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

Example: Transform the addition into multiplication

Example2: Transforming the subtraction into division

Example3: The form of Will be changed into And the same for Will be

Solve it yourself!

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: let Divide each side by Take the base-c logarithm of each side Power rule

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:

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

REEM

The Nature of Logarithm Is the logarithm to the base e, where e is an irrational constant approximately equal to

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.

For Example Property of Logarithms: Definition of Logarithm Simplify 8 = x- 7x Write quadratic equation in standard form 0 = x- 7x – 8 Solve by factoring 0 = (x – 8)(x + 1) x – 8 = 0 or x + 1 = 0 x = 8 or x = -1

Substitute the solution –1 for x Substitute the solution 8 for x Subtract The number -1 does not check, since negative numbers do not have logarithms Because ; because = 3 3=3

Proof that d/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

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..

(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: Fractions 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 Y= LOG 2(X)X log2(0.125) = – log2(0.5) = –1 0.5 log2(1) = 0 1 log2(2) = 1 2 log2(4) = 2 4 log2(8) = 3 8

EXAMPLE two: Graph First change ln into logarithm form: Loge (x) Then change to exponential form: X= e power y..Now draw you T-chart Y= loge (x)X

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 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

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+15 xx= -11, which can never be real TTherefore, No Solution

Compare between logarithm and exponential graphs:

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

y = loga x y =- loga x

y = log2 (-x). Y=loga(x+2)

y = loga (x-2)

y = logax +2 y = loga x -2

THE END