ASSIGNMENT

SCHOOL MATHEMATICS 2

LOGARITHMS

Created by: y Rindra Ayu Lovenidiana (103174016) y y Agustiana Z.A. (103174035)

Widya Adian Putra (103174060)

INTERNATIONAL MATHEMATICS EDUCATION 2010 MATHEMATICS DEPARTEMENT FACULTY OF MATHEMATICS AND SCIENCES SURABAYA STATES UNIVERSITY 2012

Before we discuss about logarithm, we will review about exponents and roots form.

A. INTEGERS EXPONENTS1. Positive Integers Exponentsa. Definition of Positive Integers Exponents

If a is real number and n is positive integers so an (called a to the power of n), it means the product of n factors that each factors is a. Therefore, positive integers exponents generally can be formed :

With a = base; n = exponents; an = integers exponents There are some properties of positive integers exponents which have learned in ninth grade. Those properties is like as below. 1) The Multiplication Properties of Numbers For a R and m, n integers positive, such that :

Proof :

Based on definition

Its proven. 2) The Division Properties of Exponents Numbers For a R, a 0 and m, n integers positive that satisfy m > n.

Proof :

Its proven.

3) The Power Properties of Exponents Numbers For a R dan m, n positive integers, such that :

Proof:

Its proven. 4) The Power Properties of Numbers Multiplication For a,b R and n positive integers, such that:

Proof:

Its proven.

5) The Power Properties of Numbers Division For a, b R, b 0 and n positive integers, such that:

Proof:

Its proven.

2. Zero and Negative Integers Exponents a. Zero Exponents For a R and a 0, so

Proof:

(division properties of exponents numbers)

=1

Its proven. Note : 00 is undefined. Because Undefined b. Negative Integers Exponents For a R and a 0 defined as: = = =

This definition comes from Suppose : = : = = = =

B. ROOTS FORM1. Irrational Numbers Concept Irrational Numbers is defined as the number which can not be stated in propotional , with a, b Z and b{0. Meanwhile, rational numbers is and b{0. The example or irrational numbers : y = 3,141592 ... y e = 2,718281 ... = 1,414213 ... y defined as the number which can be stated in propotional , with a, b

Z

y = 2, 6457... The example of rational numbers : y y y = 3,0000...

2. Roots Form The number in roots form, there are three parts must be known, roots form symbol, radican, and index. Generally, roots form can be witten as :

(

called n root of a)

With called roots form (radical), is called roots form symbol, n is called indeks (power of root), a is called radican (the number below root ), a is positive real number for n is natural number, meanwhile for n is odd number, a can be negative real number. a) Kind of Roots Form 1) The Roots of Its Namesake A roots form is called the roots of its namesake if the index are the same. Example : a. b. 2) Similar Roots A roots form is called similar roots if the index and radican are the same. Example : all of them have index 3, its radican 2.

b) Roots Properties For a, b real number, n is natural number, such that:

2. 3.

3. Unreal Exponents Exponents numbers with zero, negative integers, and fraction power are called unreal exponents. Meanwhile, exponents numbers with positive integers is called real exponents. For any value a with a{0, m is integer number, n is natural number and nu2, such that :

Number and exponents.

are called unreal

4. Properties of Unreal Exponents Operation For a, b R with a, b { 0, and p, q is rational number, such that satisfy this unreal exponents properties :

{ {

c)

Rationalizing the Denominator in Roots Form A rational of roots on its denominator; such as : can be simplified if we move roots form from the denominator. This process is called rationalizing the denominator. Rationalizing means changing roots form on the denominator become rational number form, then that number can be stated in the simple form. A fraction form that contain number in root form is called simple, if satisfied : 1. Every numbers which its root form in simple form. 2. There is no roots form in the denominator if that number is fraction. How to rationalize number : 1) Fraction in form Roots form with b{0 can be rationalized its denominator with multiply such that :

that fraction with

2) Fraction in form To simplify fraction form or with multiply that fraction with is

adjacency form from the donominator. Adjacency form of . Viceversa, adjacency form of is , such that :

3) Fraction in form To simplify denominator from roots form or , with multiply

that fraction with adjacency form from its denominator. Adjacency form of is . Viceversa, adjacency form of is such that :

4) Simplfy roots form Form requirements Proof :2

and

can be changed become form . 2

with

2

2

2 Therefore,

To get understanding better, pay attention to this example : 1. Simplify this exponents form Solution !

Using

(The multiplication properties of numbers)

Then, using the division properties of numbers

So, the solution is 2. Given x=64 and y=8. Determine the value of !

Make the base are the same

C. LOGARITHMS1. Definition Logarithms is the inverse from the exponential. That is find out the exponents from the base, then, the result is suitable with the given. The definition of logarithms : Let a is positive integers (a > 0) and g is positive integers that is not equals to 1 ( 0 < g < 1 or g > 1) Then we get : g Log x = a If and only if gx = a With : g = base, g > 0 ; a = numerous , a > 0 x = the result of logarithms, can be positive, negative or zero Example : 1. State this following logarithms in the exponents formed. a. 3Log 9 = 2 b. 2Log 32 = 2p Solution By using the definition, we get : a. 3Log 9 = 2 32 = 9 b. 2Log 32 = 2p 22p = 32 2. State this following exponents I the logarithms formed. a. 52 = 25 b. 60 = 1

Solution By using the definition, we get : a. 52 = 25 5Log 25 = 2 b. 60 = 1 6Log 1 = 0 2. The Properties of Logarithms a. Properties 1 For a > 0 , and a 1, satisfy : a Log a = 1, aLog 1 = 0, Log 10 = 1 Proof : y Every numbers if to the power 1, then the result is itself. Then, a1 = a aLog a =1 y Every numbers that is not equals to zero, if to the power of 0 then the result is 1. Then, a0 = 1 aLog 1 = 0 y Log 10 is the logarithms form with the base and numerous 10. Then, Log 10 = 1 b. Properties 2 For a > 0, a 1, x > 0 and y > 0 , a, x, and y R satisfy : a Log x + aLog y = aLog xy Proof : a Log x = n an = x a Log y = m am = y a Log xy = p ap = xy from the exponents formed, we get : xy = anam xy = an+m ap = an+m p = n + m then, n = aLog x, m = aLog y , and p = aLog xy, So, its proven that aLog x + a Log y = aLog xy c. Properties 3 For a > 0, a 1, x > 0 and y > 0 , a, x, and y R satisfy :a

Log x - aLog y = aLog

Proof : a Log x = n an = x a Log y = m am = y a Log = p ap = from the exponents formed, we get : ap = an m p=nm So, its proven that aLog x - aLog y = aLog d. Properties 4 For a > 0, a 1, , a, n, and x R satisfy : a Log xn = n a Log x Proof :

a

Log xn = aLog

Log xn = aLog x + aLog x + a Log x +.. aLog x = n a Log x e. Properties 5 For a, m > 0 and a, m, n, x R, satisfy :

a

Proof :

From the exponents form, we get :

Then, its proven that :

Example : 1. Simplify : a. 2Log 4 + 2Log 8 b. 5Log 3 + 5Log 50 Solution : Log 4 + 2Log 8 = 2Log (4 x 8) ... (by using properties 2) = 2Log 32 = 5 b. 2Log 16+ 2Log 4 = 2Log (16 x 4) .... (by using properties 2) = 2Log 64 = 6 2. Simplify : a. Log 0,04 Log 4 b. 7Log 217 7Log 31 = a. Solution : a. Log 0,04 Log 4 = Log b. 7Log 217 7Log 31 = 7Log properties 3) 3. Find x of the following logarithms: .... (by using properties 3) = 7Log 7 = 1 .... (by using2

= Log 0,01 = -2

Solution :

= =

...... (by using properties 4)

= Log 2 + Log 9 Log 3 = Log .... (by using properties 2 and 3) = Log 6 Log x = Log 6 x =6 f. Properties 6 For a, p > 0, and a, p 0 , a, p, x R, satisfy :

Proof : a Log x = n x = an Log x = Log an Log x = n Log a na

=

Log x =

If p = x, then,a a

Log x = Log x =

g. Properties 7 For a > 0, x > 0, y > 0, a, x, and y R, satisfy : a Log x . xLog y = aLog y Proof : a Log x = p ap = x x Log y = q xq = y From the exponents form, we get : y = xq y = (ap)q y = apq a Log y = aLog apq a Log y = pq aLog a a Log y = pq a Log y = aLog x . xLog y = q h. Properties 8 For a > 0, a and x R, satisfy :

Proof: a Log x = n an = x x = an x =

=x

i. Properties 9 For a > 0 and a, x R, satisfy :

Proof : n aLog x = p aLog xn = p xn = ap xn =

Example : 1. If 2Log 3 = a and 3Log 5 = b. State this following logarithms in a or b. a. 2Log 5 b. 6Log 15 Solution :2

Log 3 = a log 2 = Log 3

3

Log 5 = b log 5 = b Log 32

a. b.

Log 5 = Log 15 =

... (using properties 6)

6

... (using properties 6)

2. Simplify : a. b. Solution : a. = 3Log = = = b. = = 22 + 32 3 = 4 + 9 -3 = 10 + 2Log 33 +

D. EXPONENTS FUNCTION1. The Properties of Exponents Function To solve exponents equation, it is to be better if we remember again about the properties of exponents function that we have learned before. If a, b R, a 0, m and n are rational number. You can see some properties of exponents function below :

2. Exponents equation Exponents equation is an equation that exponents and the base contain of variable. You can see the example above : y is the exponents equation that the exponents contain

of variable x. y is the exponents equation that the exponents and it base contain of variable y. y is the exponents equation that the exponents contain of variable t. There are some form in exponents equation, you can see it below : a) To find x, firstly we should change the base, then , why we change 1 to be ? Because we know that every number that have power zero, the result is one. From above we get, f(x)=0, for , a>0 Generally, it can be written as :If

, with exponents equation is

, , then the solution set from the that satisfy

b) If c) If , a>0 dan , then f(x) = g(x) , a>o and a , then f(x) = m

Exponents equation , it means If only have solution if .

If If g(x) = h(x) f(x) = 1

then the solution are :

then the solution are :

f(x) = 0, with g(x) and h(x) are positive f(x) = -1, with both of them are even or both of them are odd.

f) First, assume . From this assumption, we get . Then you will get the value and you can substitute the value of y in the assumption . Then we can find the value of x. 3. Exponents inequality You have learned about the properties of exponents function. for a > 1, is ascending function. It means, for every is applied < if and only if < . for 0 < a < 1, is descending function. It means, for every is applied < if only if > . These properties is used for solve exponents inequality .

E. EQUATION AND INEQUALITY LOGARITHMS1. The Properties of Logarithms Function We have learned about logarithmss properties before. In general, we can write the logarithms function as below : with a > 0 and a 1

2. Logarithms function Logarithms function is an equation that the variable as a numerus or as a base from a logarithms. You can see the example below : is an logarithms equation that the numerus is contain of variable x. is logarithms equation that the numerus contain of variable m. is logarithms equation that the base contain of variable x. is logarithms equation that the numerus and the base contain of variable t. There are some form of logarithms equation, as follows: If

If c) If d) If

e)

first, assume

. From this assumption we get . Then you will get the value of y. Substitute the value of y in . After that, you will get the value of x.

3. Logarithms inequality You have learned about the properties of exponents function, as follows: For a > 1, is ascending function. It means, for every

is applied for 0 < a < 1, every is applied

is descending function. It means, for