STANDARD FORM, INDEX AND LOGARITHMS BA101/CHAPTER2 1 Prepared By : Azmanira Muhamed CHAPTER 2: STANDARD FORM, INDEX AND LOGARITHMS 1.0 INTRODUCTION Understanding index and logarithms is important because it is used widely in research, science, finance and engineering. For example, it can be used to show how substances are formed, multiply and decay in the natural world. Index Expressions Figure 1.1 shows our planet Earth orbiting the Sun. The distance between the Earth and the Sun is about 93 million kilometers. You can write this number as 93,000,000 km. Figure 1.1: Distance between Earth and Sun This number is obviously long to write and hard to read. You can also write this number as 9.3 x 10 7 km. Numbers written this way is called indices. Usually only numbers that are too small and too big are stated in index form. Referring to the number 9.3 x 10 7 , the number 10 is called the base and the number 7 is called the index.
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STANDARD FORM, INDEX AND LOGARITHMS BA101/CHAPTER2
1 Prepared By : Azmanira Muhamed
CHAPTER 2: STANDARD FORM, INDEX AND LOGARITHMS
1.0 INTRODUCTION
Understanding index and logarithms is important because it is used widely in research,
science, finance and engineering. For example, it can be used to show how substances are
formed, multiply and decay in the natural world.
Index Expressions
Figure 1.1 shows our planet Earth orbiting the Sun. The distance between the Earth and the
Sun is about 93 million kilometers. You can write this number as 93,000,000 km.
Figure 1.1: Distance between Earth and Sun
This number is obviously long to write and hard to read. You can also write this number as
9.3 x 107km. Numbers written this way is called indices. Usually only numbers that are too
small and too big are stated in index form. Referring to the number 9.3 x 107, the number 10
is called the base and the number 7 is called the index.
STANDARD FORM, INDEX AND LOGARITHMS BA101/CHAPTER2
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In general, an means a multiplied by itself n times.
An index expression is in simple form if there is:
No repeating base
No negative index
For example
x2y5 x 4 , a2 b –6 and23p
can be simplified as
x6y5, 6
2
b
a and
9p
1012
= 10 x 10 …………10 .
12 times
base
index
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Activity 1.0
TRY THESE QUESTIONS!
1. Rewrite these numbers in index form:
i. a. 2000000 b. 138000000000
ii. c. 0.00082 d. 0.000000015
2. The rate of reproduction of a particular insect is about 1430000000 a month. Write this in
index.
3. First, express 400 kilometers in centimeters. Next, state this in index.
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FEEDBACK for Activity 1.0
1. a. 2 x 106 b. 1.38 x 1011
a. c. 8.2 x 10-4 d. 1.5 x 10-8
2. 1.43 x 109
3. 40 000 000 cm, 4 x 107 cm
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1.1 INDEX RULE
There will be times when you need to add, subtract, multiply or divide two or more index
numbers. The rules provided in Table 1.1 are most useful. Study it carefully.
Given that a, b, m and n are real numbers.
Rule Statements
1. Multiplication
am x an = a m + n
2. Division
am an = a m - n
3. Power
i. ( a m ) n = a mn
ii. (ab)n = an b n
iii. n
nn
b
a
b
a
; b 0
4. Negative Index i. a –n =
na
1 ; a 0
ii. m
n
n
m
a
b
b
a
; a 0 and b 0
5. Zero Index
a0 = 1 ; a 0
6. Fraction Index
n mn
m
aa
Table 1.1: Rules of Index Operations
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Example 1.1
Simplify the following expressions.
a. 34 x 35 b. 85 82
c. (48 )3 d. ( 25 ) -3
e. 70 f. 3
2
27
SOLUTION:
Expressions Index Rule Used Solutions
a. 34 x 35 am x an = a m + n 34 + 5 = 3 9
b. 85 82 am an = a m - n 85 – 2 = 8 3
c. ( 48 )3 i. ( a m ) n = a mn 424
d. ( 25 ) -3
i. ( a m ) n = a mn
ii. a –n = na
1 ; a 0
( 2 5 ) -3 = 2 -15
2 –15 = 152
1
e. 70 a0 = 1 ; a 0 1
f. 3
2
27 n mn
m
aa 92727 3 23
2
Example 1.2
A sum of RM10,000 is saved in a bank at 8% interest compounded monthly. The total sum J
after t years is given as, J =
t12
12
08.0110000
. What is the total sum after
a. 6 months b. 5 years
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SOLUTION
a. 6 months = 0.5 year. Therefore t = 0.5.
J =
6
12
08.0110000
= 10406.73
b. t = 5.
J =
)5(12
12
08.0110000
=14898.46
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Activity 1.1
TEST YOURSELF BEFORE YOU MOVE ON TO THE NEXT SECTION……..!
1 Simplify
a. a5 x a 6 b. 3 x 3 8 x 3-4
c. z 2 y5 x y –2 d. 4n x 16 2n x 32 –2n x 8 -n
2. Simplify
a. m 12 m 3 b. 2 8 2 4
c. z 7 x z 6 z 5 d. 25 n 5 –2n x 125 2n
3 Simplify
a. ( x 3 ) 5 b. ( 3x 4) 2
c. ( 2x2 y 3 z )5 d. ( 10 3 ) 4
e. 3 ( ab 2 ) –4 f. ( 2m 2 ) ( 4n )3
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Feedback for Activity 1.1
4. a. a11 b. 35
c. z2 y 3 d. 2 –3n
5. a. m 9 b. 24
c. z 8 d. 5 10n
6. a. x 15 b. 9x 8
c. 32 x 10 y 15 z 5 d. 1012
e. 84
3
ba f. 3
2
32n
m
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1.2 LOGARITHMIC EXPRESSIONS
In this section, you will learn about the relationship between indices and logarithms. Let’s
now consider the reproduction rate of amoeba, a single cell living organism that reproduces
by replicating itself. If one new amoeba needs one day to replicate itself into 2 amoebas,
there will be 4 amoebas after 2 days, 8 amoebas after 3 days, and so on.
Time in days(x) 0 1 2 3 4 5 6 7
Number of amoeba (y) 1 2 4 8 16 32 64 128
The index equation y = 2x can be used to represent the rate of reproduction of this amoeba.
Conversely, this equation can also be written as x = log2 y , a logarithmic equation.
log2 y is read as log of
y to the base 2
log is a short form of logarithm
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Note: Observe that the base is still the same after changing index form to logarithmic form.
Generally, if a is a positive number and y = a x, then x is equals to the logarithm of y to
the base a.
Equation 1.1
Example 1.3
Rewrite the following numbers in logarithmic form, (base given).
a. 100 base10 b. 64 base 4
c. 64 base 2 d. 125 base 5
e. 81 base 3
Sample solutions
a. If 100 = 10 2, then 2 = log 10 100
b. If 64 = 4 3, then 3 = log 4 64
c. If 64 = 2 6, then 6 = ?
Index Form Logarithmic Form
index
y = ax x = log a y
base
If y = a x , then x = log a y
If x = log a y , then y = a x
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d. If 125 = 5 3, then ? = log 5 125
e. If 81 = 3 4, then 4 = log ? 81
Example 1.4
Determine these values:
a. log 7 49 b. log 2 0.5 c. log 9 3 d. log 10 0.001
Solution:
a. Assuming log 7 49 = x
then 49 = 7x
72 = 7x
therefore x = 2
b. Assuming log 2 0.5 = x
then 0.5 = 2 x
0.5 = ½ = 2 – 1 = 2 x
therefore x = -1
c. Assuming log 9 3 = x
then 3 = 9 x
3 = ( 3 2 )x
31 = 32x
therefore 1 = 2x
½ = x or x = 0.5
d. Assuming log 10 0.001 = x
then 0.001 = 10 x
x1010
10
1
1000
1 3
3
therefore x = -3
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Activity 1.2
TEST YOURSELF BEFORE YOU MOVE ON TO THE NEXT SECTION……..!
1. Given that 32 = 25, determine the value of log 2 32
2. Given that 1/8 = 2 –3 , determine the value of log 2 1/8
3. Given that 8 = 64 , determine the value of log 64 8
4. Given that 0.001 = 10 –3 , determine the value of log 10 0.001
5. Calculate the value of
a. log 3 81 b. log 7 343
c. log 8 4 d. log 27 9
6. Convert the following into logarithmic form
a. 32 = 25 b. 50 = 10 1.699
c. a = x2 d. x-3 = 0.3
7. Find the value of x given that
a. log 3 1 = x b. log 7 49 = x
c. log 10 0.001 = x d. log 5 25 = x
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Feedback for Activity 1.2
1. 5
2. –3
3. ½
4. –3
5. a. x = 4 b. 3
c. x = 2/3 d. 2/3
6. a. log 2 32 = 5 b. log 10 50 = 1.699
c. log x a = 2 d. log x 0.3 = -3
7. a. x = 0 b. x = 2
c. x = -3 d. x = 2
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1.3 LOGARITHM RULE
As in index, you will need to perform basic algebra operations on numbers in
logarithm. Table 1.2 shows the logarithm rule.
Table 1.2: Rules of Logarithm Operations
Example 1.5
Write the following expressions as addition or subtraction of logarithms.
a. log a x 2 y 3 b. log a 3 b 3/2
c. 2100
1log
b c. )(log
3
c
ab
Solution:
a. log a x 2 y 3 = log a x2 + log a y 3
= 2 log a x + 3 log a y
b. log a 3 b 3/2 = log a 3 + log b 3/2
= 3 log a + 2
3log b
Assume M and N are positive real numbers
1. log a MN = log a M + log a N
2. log a M/ N = log a M – log a N
3. log a (M) c = c log a M
4. log a a = 1
5. log a a0 = 0 log a a = 0
6. log N M = N
M
a
a
log
log ( to convert base N to base a)
7. a log
a N = N
8. log101 = log a (for base 10 only)
RULE 1 and 3
STANDARD FORM, INDEX AND LOGARITHMS BA101/CHAPTER2
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c. 2100
1log
b = log 1 – ( log 100 + log b2 )
= 0 – log 100 – 2 log b
= - ( log 100 + 2 log b )
= -2 (1 + log b )
d. )(log3
c
ab=
2/13
log
c
ab= ½ ( log ab3 – log c )
= ½ log ab3 – ½ log c
= ½ log a + ½ log b3 – ½ log c
= ½ log a + 2
3log b – ½ log c
Example 1.6
Rewrite the expressions below as a single logarithm
a. log 2 + log 3 b. log 4 + 2 log 3 – log 6
c. 2 log x + 3 log y – log z d. ½ log a + log a2 b – 2 log ab