Matematika 1

1. CHAPTER 1 Akar, Pangkat dan Logaritma Matematika 1 FOR SENIOR HIGH SCHOOL

2. AEksponen • Pangkat Bulat Positif a. Definisi Pangkat Bulat Positif The positive integer exponent is a repeated multiplication of the same positive number. Example : 4 x 4 x 4 x 4 x 4 is written as 45 and read as four to the power five.

3. Let a is a real number ( a Є R ) and n is positive integer greater than 1 ( n Є A, n > 1 ), then the power an read as a to the power n ) says that n copies of a are multiplied together. Mathematically, it is written as: an = a x a x a x … x a x a Comprised of the same number n The number a is called base, while the number n is called power or exponent

4. b. Properties of the Number with the Positive Integer Exponents Properties Let m, n Є A and a , b Є R , then the properties below are hold. 1. Multiplication property am . an = am + n 2. Division property = am – n , with a≠0 and m>n 3. Exponentiation property (am)n = amn 4. Multiplication and exponentiation property ( a.b ) m = am . bm 5. Division and exponentiation property = , with b≠0

5. Example : • ( 32 )4 = 32.4 = 38 • = a4-2 = a2

6. 2. Negative Integer and Zero Exponents • Negative Integer Components Definition : For every a Є R, a ≠ 0, and the positive integers n , then a-n = or an = Example : Observe that a4 : a6 = a4-6=a-2 or So, a-2 =

7. b. Zero Exponents Definition : For every a Є R and a ≠ 0 then ao = 1 Example : Simplify and state each of the following expressions in their positive integer exponents ! a. 2p3q-4 b. 4p2p-3 Answers : a. b.

8. 3. Expressing the Number in Standard Form (Scientific Notation) A number N expressed in scientific notation is the product of an arbitrary number a ( between 1 and 10 ) and exponent with base 10. Mathematically, N = a x 10n, where 1 ≤ a < 10 and n is integer. Example : A number 0.078 can be expressed in standard form by shifting the point two places to the right and multiplying it with 10-2, thus 0.078 becomes 7.8. Multiply with 102 x 10-2, so 0.078 = ( 0.078 x 102 ) x 10-2 = 7.8 x 10-2

9. B Root • Definition of Rational and Irational Numbers Rational numbers are numbers that can be expressed as fraction , where a and b are integers and b ≠ 0. Irational numbers are numbers that cannot be expressed as fraction , where a and b are integers and b ≠ 0. Note : is not irrational number, because =2

10. 2. Definition of Root Roots are numbers in the root symbol which cannot produce rational numbers. Examples : a. =0.8 ( not root notation ) b. = 1.26491 …. ( root notation )

11. 3. Algebra Operation of the Roots • Addition and Subtraction of the Roots Property : • For Every a, b Є R and c Є R+ , then Examples :

12. b. Multiplication of Roots Properties For every a, b, c, d Є R and c, d ≥ 0, then 1. 2. 3. Example :

13. 4. Simplifying the Form of Often you have problems in the form of . To simplify the problems, you may manipulate them to the form of , where p > q. Study the following problem! Thus, we obtain the following expression. Applying the same method, we get the form below.

14. 5. Rationalizing the Denominator of a Fraction a. Fractions in Form of A fraction in form of can be rationalized by multiplying the fraction number with b. Fraction in Form of A fraction in form of can be rationalized by multiplying the fraction number with c. Fraction in Form of A fraction in form of can be rationalized by multiplying the fraction number with

15. 6. Fraction Exponents a. Fraction Exponents in Form of Properties • If a is real number, m is integer, n is natural number and n ≥ 2, then • If a > 0, then we can apply the property above for both odd and even numbers m and n. • If a < 0, then we cannot apply the property above for both odd and even numbers m and n. Example :

16. b. Properties of Rational Exponents If a, b Є R, a ≠ 0, b ≠ 0 with m and n are rational numbers, then we have 1. 6. 2. 7. 3. 8. 4. 9. 5. Example:

17. c. Solving Simple Rational Exponent Equation with the Same Base An Equation involving variable exponents is defined as exponent equation. The examples of exponent equations are: The examples above can be written as . To determine the value of x , we may use the following property: Property If a where a Є R (a ≠ 0 and a ≠ 1 ) then f(x) = p

18. Example : Thus, x = -3 or x = 1

19. C Logarithm 1. Definition of Logarithm If a > 0, a ≠ 1, and alogb = x then b=ax where a is called base, b is called numerous, and x is product of logarithm. Example :

20. 2. Determining the Logarithm of the Numbers • Determining Logarithm Values Using Logarithm Table. The procedures of reading the logarithm table above are as follows: 1 ) If you want to determine log 1.94, the first step is finding that two first numbers in column N, that is 19. 2 ) The next step is finding decimal ( mantis ) in the row 19 and column 4 ( the sixth column), you will get value 288. 3 ) Since the value 1.94 is between 1 and 10, so the integer of 1.94 is 0. Thus, we have log 1.94 = 0.288.

21. b. Determining Logaritm Values Using a Calculator The steps of operating calculator to determine the logarithm value of numbers are as follows. 1 ) Press the log button. 2 ) Type a number of which you need to find the logarithm value. 3 ) Press the buttons of = Press the log button Type a number Press the log button Result

22. 3. The Properties of Logarithm Properties If a, b, c are positive real numbers and a ≠ 1, then the following properties are hold. 1. 5. 2. 6. 3. 7. 4. 8.

23. Example : • If 2log 3 = a and 3log 7 = b , express each of the following problems in terms of a and b. a. 2log 98 b. 8log 49 Answers : a. 2log 98 = 2log ( 2 x 49 ) =2log 2 + 2log 49 =1 + 2log 72 = 1+ 2.2log 7 =1 + 2. 2log 3. 3log 7 =1 + 2ab

24. 4. Determining the Logarithm of the Numbers Greater than 10 and Numbers between 0 and 1 To determine the logarithm of the number greater than 10 and numbers between 0 and 1, you can apply the properties as you have already learned. In the first step, you need to convert that numbers in standard form where 1≤ a < 10 and n is integer. Then, you may use some of the logarithm properties, as the following. log ( a x 10n ) = log a + log 10n = log a + n log 10 = n + log a

25. Example : Solve the following logarithms ( number between 0 and 1 ) • Log 0.532 b. log 53.2 Answers : • Log 0.532 = log 5.32 x 10-1 = log 5.32 – 1 = 0.726 – 1 = -0.274 b. log 53.2 = log ( 5.32 x 101 ) = log 5.32 + log 101 =0.726 + 1

26. 5. Determining Antilogarithm of a Number Determining the antilogarithm of a number means finding a number which is given its logarithm value using antilogarithm table. To determine antilogarithm of a number, study the following example. Determine the numbers which satisfy the following logarithm! a. 0.125 Answer: a. Antilog 0.125 = 1.33 From the antilogarithm table, find the two first decimals in the most left column ( column x ), that is 12, then draw a line horizontally from that number to the right until intersect the column which indicates number 5, so you obtain 1.33. Because the integer is 0, thus the antilog 0.125 = 1.33

27. 6. The Application of Logarithm in Calculation • Using Logarithm in Multiplication and Division The logarithm properties usually used in multiplication and division are as follows. • Log ( a x b ) = log a + log b • Log = log a – log b Example : • 4.28 x 15.62 = … Answer : Let p = 4.28 x 15.62 Log p = log (4.28 x 15.62 ) Log p = log 4.28 + log 15.62 Log p = 0.631 + ( 0.193 + 1 ) Log p = 1.824 Log p = 0.824 + 1 p = antilog 0.824 + antilog 1 p = 6.67 x 10 p = 66.7

28. Using Logarithm in Exponentiation and Root The Logarithm properties usually used in exponentiation and root are as follows Example : Calculate by using logarithm! Answer : Let p = Let p = log Log p = x log 51.2 Log p = x 1.709 Log p = 0.8545 p = antilog 0.8545 p = 7.16