CHAPTER 1
Akar, Pangkat dan Logaritma
Matematika 1
FOR SENIOR HIGH SCHOOL
A Eksponen
1. 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.
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
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
b≠0
=
, with
Example :
• ( 32 )4 = 32.4 = 38
•
= a4-2
= a2
2. Negative Integer and Zero Exponents
a. 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 =
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.
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
B Root
1. 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
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 )
3. Algebra Operation of the Roots
a. Addition and Subtraction of the Roots
Property :
• For Every a, b Є R and c Є R+ , then
Examples :
b. Multiplication of Roots
Properties
For every a, b, c, d Є R and c, d ≥ 0, then
1.
2.
3.
Example :
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.
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
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 :
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.
5.
Example:
9.
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
Example :
Thus, x = -3 or x = 1
C Logarithm
1. Definition of Logarithm
If a > 0, a ≠ 1, and alog b = x then b=ax where a is called
base, b is called numerous, and x is product of logarithm.
Example :
2. Determining the Logarithm of the Numbers
a. Determining
Table.
Logarithm
Values
Using
Logarithm
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.
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
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.
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
b. 8log 49 =
=
2
=
2
log 72
log 7
log 3. 3log 7
= ab
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
Example :
Solve the following logarithms ( number between 0 and 1 )
a. Log 0.532
b. log 53.2
Answers :
a. 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
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
6. The Application of Logarithm in Calculation
a. Using Logarithm in Multiplication and Division
The logarithm properties usually used in multiplication and
division are as follows.
1.
Log ( a x b ) = log a + log b
2.
Log
= log a – log b
Example :
1.
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
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