Download Introduction to Logarithms

Introduction to Logarithms
Logarithms are commonly credited to a Scottish mathematician named John Napier who
constructed a table of values that allowed multiplications to be performed by addition of the values
from the table.
Logarithms are used in many situations such as:
(a)
Logarithmic Scales: the most common example of these are pH, sound and earthquake
intensity.
(b)
Logarithm Laws are used in Psychology, Music and other fields of study
(c)
In Mathematical Modelling: Logarithms can be used to assist in determining the equation
between variables.
Logarithms were used by most high-school students for calculations prior to scientific calculators
being used. This involved using a mathematical table book containing logarithms. Slide rules were
also used prior to the introduction of scientific calculators. The design of this device was based on
a Logarithmic scale rather than a linear scale.
There is a strong link between numbers written in exponential form and logarithms, so before
starting Logs, let’s review some concepts of exponents (Indices) and exponent rules.
The Language of Exponents
a n can be written in expanded form as:
a n = a × a × a × a × a............ × a [ for n factors ]
The power
The power
a n consists of a base a and an exponent (or index) n.
Base
n
Exponent (or index)
a ‘The number of times it is repeated.’
‘The number being repeated.’
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Examples:
3^5= 243
3W5é 243
35 = 3 × 3 × 3 × 3 × 3 = 243
Multiplication or division of powers with the same base can be simplified using the Product and
Quotient Rules.
The Product Rule
am × an =
a m+n
When multiplying two powers with the same
base, add the exponents.
32 × 34 = 36 = 729
4−1 × 45 × 42 = 4−1+5+ 2 = 46 = 4096
b 2 × b 7 = b 2+ 7 = b9
The Quotient Rule
am ÷ an =
a m−n
When dividing two powers with the same
base, subtract the exponents.
37 ÷ 32 = 37 − 2 = 35 = 243
42 × 44 ÷ 43 = 42+ 4−3 = 43 = 64
e6 ÷ e 4 = e6− 4 = e 2
The zero exponent rules can also be used to simplify exponents.
The Zero Exponent Rule
0
a =1
A power with a zero exponent is equal to 1.
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30 = 1
1137500 = 1
e0 = 1
0
( 4x) = 1
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The power rule can help simplify when there is a power to a power.
The Power Rule
(a )
m n
=a
Two more useful Power Rules are:
m m
(ab) = a b
m
or
m
a
a
  = m
b
b
m m
a b = (ab)
m
or
2×4
0 6
When a power is raised to a power, multiply
the exponents.
m
( 3 )= 3 = 3= 6561
( 5 )= 5 = 5= 1
(b=
) b= b
2 4
m×n
a
a
= 
m
b
b
m
m
0×6
2 7
3
2 3
0
2×7
( 3a )=
( 5a )
8
14
33 × a 3
=
53 × a 2×3 =
125a 6
4
4
256
4 4
=
=
 
4
a
a a
The negative exponent rule is useful when a power with a negative exponent needs to be
expressed with a positive exponent.
The Negative Exponent Rule
a
−m
1
1
m
or
=
a
am
a−m
1
=
32
1
1
4−=
=
41
1
b −7 = 7
b
2
3−=
Take the reciprocal and change the sign of the
exponent.
1
9
1
4
4
=4 × a 3 =4a 3
−3
a
The fraction exponent rule establishes the link between fractional exponents and roots.
The Fractional Exponent Rule
1
n
1
2
3=
a = a
m
n
n
2
3
( a)
n
=
82 4
3
4
p =
m
for any fraction
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−
1
4
b=
3
3
=
8
for unit fractions, or
a = n a m or
=
3
2
4
1
=
1
b4
p3
1
4
b
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Exponent Functions found on a Scientific Calculator
Function
Appearance of Key
Example
Square 32
d
3d= 9
Cube 23
D
2D= 8
Any exponent 35
for W
16
s or s
Square root
3^5= 243
3W5é 243
s16= 4
qD343= 7
Cube root
3
343
S or S
Generally qDgives S or S
(although this varies between different
makes and models)
10qf200= 1.699
10q W200p 1.699
Any root
10
200
F or
x
Generally qfgives F or
qW gives
x
(although this varies between different
makes and models)
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