Just For Mathematics

Properties of Logarithms
Logarithms is a type of mathematics which related to power and root. That the power of root are used for a basic to learn it. Therefore, we should be familiar and understand abaut that. For example : ten to the second power is one hundreds, the square root of onen hundred and the fourty four is twelve. Let’s start to talk abaut the logarithms and its properties. This explanation I get from browsing in the internet. There is definition Log b X= y
It means that b to the y power equals y. Besides that there is a agreement abaut writing log 10 X. It can be written just log X and log e X equals ln x. It is called as natural logarithms. Then, let’s start to do the exercise to find a solving in logarithms.
If log 10 100 is x. Find the value of x!
• First, make a equation, ten to the x power is one hundred.
• Remember that one hundred is ten to the second power.
• From statement above we can conclude that the value of x is two because ten to the second power is one hunderd. Therefore, two is right answer.
Next question, if log 2 x is three. Find the value of x!
Write that two to the three is x. Therefore x is eight. In logarithms, there are some properties like that:
Log b (M times N) equals log b M and log b N
Log b (M/N) equals log b M minus log b N.
Log b (x^n) equals n times log b x
Let’s practise it. If log 3 (x^3(y+1)/z^3). How find the solution?
First, log 3 (x^3(y+1) minus for log log 3 z.
Second, log 3 x^3 and log 3 (y+1) minus for 3 log 3 z.
Next, 3 log 3 (x^3(y+1)/z^3)= 3 log3 x + log 3 (Y+1) – 3log3 z.
To get more complete explanation and exercise, we can search in the internet or read a bool about logarithms because it just explain the concept or the principle of properties of logarithms




Common Factor and Grouping
( Greatest Common Factors)

I’ m sure that most student know about GCD and LCM. GCD means the greatest common divisor while LCM means the least common multiple. We got them when we studied in Elementary school. In this section, explain of greatest common factors or written GCF in Indonesia “ Faktor Persekutuan Terbesar.
The objective of greatest common factor are :
 Find the greatest common factor (GCF) of numbers
 Find the GCF of them
 Factor out the greatest common facto
Factor a four term expression by grouping and the getting started are product and factor. For example:
20=2x2x5, two and five are prime numbers.
Next, how we got GCF? Finding the greatest common of number are :
 The greatest common factor at a list of integers in the largest common factor of the integers in the list.
45=3x3x5=3^2x5
60=2^2x3x5
 To find the greatest common factor (GCF), choose prime factors with the smallest exponent and find their product.
For example : find GCF of thirty six, sixty and one hundred and eight.
Thirty six is two square times three square or 36=2^2x3^2
Sixty is two square times three times five or 60=2^2x3x5
One hundred and eight is two square times three cube or 108=2^2x3^2 so the GCF is two square times three equals twelve.
Others way:
2 36 60 108 thirty six, sixty and one hundred and eight divided by
2 18 30 54* two , the result is * then divided by two again, the result
3 9 15 27# is #. After that divided by three.
3 5 9

Function
Based on Oxford Advanced Learner’s dictionary, function is a quantity whose value depends on the varying value of others. In the statement 2x=y, y is a function of x, or functioin is an algebra statement that providers a link between two or more variable . function used to find the value of one variable. If you know the value of the others. y=2x, if you know x, you can find the value of y.
For example x is five so that the value of y is two times five equals ten. One variable appears by itself, on one side of the equation y=3x+4. A function is a codependent relationship between gx’s dan y’s without x, you can’t get y. Function related in which each element of one set is paired with one and only one, element of the second set. Besides that any numerial expression relations one number, or set of numbers, to another. There are two kinds of relations namely equations and inequalities. The sample of equation is 1+3=4 while inequality is 8>5 (eight more than five).
Specific numbers are being related to each other nonspecific numbers. Nonspecific number related to variable, namely x and y. X and y related to variables relations which have expression that contain variables, example: 2x+3=9y.
Equation and inequalities are the kind that can be used to determine just one value for one of the variable. Y=3x+4, when you substitute a value for x, you can calculate just one value for y. Y=3x+4 an equation with one variable by itself on one side, one variable is a function of whatever variable appear on the other side. Function of x, f(x)=y, y=3x+4, 3x+4 is function of x so y=f(x)=3x+4. F(x)=3x+4 is standart form, put functions that are not in standart form into standart form.
Given an equation for a function, you can calculate the function as soon as you know a value of x
g(x)=x^2-3x+2, x is five so that , g(x)=5^2-3.5+2
five to the second power minus three times five plus two is twelve.


Parallelogram
Geometri is branch of mathematics dealing with measurement and relationship of lines, angles, surfaces and solids. In geometri, we know about plane, such us: triangle, rectangle, square, rhombus and parallelogram.
Triangle has three sides and sum of the angle is one hundred and eighty degree. There are some kinds of triangle, namely: right triangle, scalene triangle and isosceles triangle. To more complete, we can browse in the internet or find the geometry book. In this section, we just talk about parallelogram. Before, I want to give definition about quadrilateral. Quadrilateral is space which has four sides. If a quadrilateral is a parallelogram, then opposite sides ara parallel.
Picture parallelogram
If ABCD is a parallelogram, then ABIIDC
ADII BC
Parallelogram has four sides, has four angles and has two pairs of parallel sides. The sum of the angle of parallelogram is three hundred and sixty degree.

Trigonometri Functions
Based on Oxford Advanced Learner’s dictionary, trigonometri is the type of mathematics that deals with the relationship between the sides and angles of triangle. Trigonometri functions only need to know values of sides to find measure of an angle figure out values of all part of a triangle. There are six basic trigonometri functions:
1. Sine
2. Cosine
3. Tangent
4. Cosecant
5. Secant
6. Cotangent
Basic trigonometri functions define six basic based on: 1) sides of a triangle, 2) Angle being measured
To make easily we learnt about trigonometri functions, let’s use a picture. It’s difficult is we don’t use picture to make us understand


OPP


Notes: OPP is side opposite theta
ADJ is side adjacent to theta
HYP is hypotenusa, the following are formula for six basic trigonometri function
Sin=OPP/HYP
Cos=ADJ/HYP
Tan=OPP/ADJ
Csc=HYP/OPP
Sec=HYP/ADJ
Cot=ADJ/HPP

For example: in this picture of triangle , find the value of sin, cos and tan!



Factoring Polynomials
Based on Oxford Advanced Learner’s dictionary, factorial means the result when you multiply a whole number by all the number below. But, in this section we don’t talk about factorial but factoring it’s different. Polynomial have many terms. The form of polynomials is a0+a1x+a2x^2+...
Now let’s learn about Factoring polynomials. Before, I want to review that factor of fiveteen is five and three. Five times equals fiveteen so five and three are the factoring of fiveteen. Next, is x minus three (x-3) a factor of x^3-7x-6? (x cube minus seven x minus six?) or we can write =(x^3-7x-6)/(x-3), how we do that?
x-3 x^3+0x^2-7x-6 x^2+3x+2
x^3-3x^2
3x^2-7x
3x^2-9x
2x-6
2x-6
0
So x-3 (x minus three) is a factor of x^3-7x-6. X^3-7x-6 divided by x-3 equals x^2+3x+2 ( it also a factor of x^3-7x-6). So we can write that x^3-7x-6= (x-3) (x^2+3x+2)
= (x-3) (x+2) (x+1)
X=3 atau x=-2 atau x=-6
The conclution
X cube minus seven x minus x has three roots, for this 3rd degree equation. Quadratic (2nd degreea) equation always have at most two roots. Ath degtee equation would have four or fewer roots. The degree at a polynomial equation always limits the number of roots. Nest, if there are long divisi for a 3rd order polynomial, the steps to find solution are : find a partial equation of x^2, by dividing x into x^3 to get x^2. Multiply x^2 by the divisor and substract the product from the diveident. Repeat the process until you either “clear it out” or reach a remainder.