Laws of Logarithms
Further Mathematics SS 1 FIRST TERM
WEEK TWO
SS1 FURTHER MATHS FIRST TERM
LOGARITHMS
CONTENTS:
 Laws of Logarithms
 Change of Base of Logarithms.
 Use of Tables (greater than one and less than one).
 Logarithmic Equations.
SUB TOPIC: LAWS OF LOGARITHMS
In the last topic indices, we learnt that p = a^{x} , e. g 1000 = 10^{3} where 3 is called the index. We can express the same in logarithms form. Log _{a }^{p} = x or log _{10} 1000 = 3
What then is logarithms?
The logarithms of a number p to base a, where a is a positive number not equal to 1 is the index to which a must be raised to give p. This shows clearly that indices and logarithms are the same.
Log_{2}8 = 3 because 8 = 2^{3}
Log_{ 3}9 = 2 because 9 = 3^{2}
Laws of Logarithms
 Log_{a}(pq) = Log_{a}p + Log_{a}q = Multiplication rule
e.g. if Log_{3}(6×5) = Log_{3}6 + Log_{3}5
 Log_{a}a = 1 e.g Log_{10}10 = 1
 Log_{a}(x/y) = Log_{a}x – Log_{a}y = Division Rule
 Log_{a}(x)^{n} = nLog_{a}x
 Log_{a}1 = 0
 Log_{a}(1/x) = Log_{a}x^{1 }= 1Log_{a}x
 If Log_{b}y = 1 then y = b
 If Log_{b}a = 1/Log_{a}b
 Log_{b }= 1/nLog_{b}x
 Log_{b}y. Log_{y}b = 1 for b and y positive and not equal to 1
 Log _{b}y = Log _{b}y / Log _{a}b
Examples:
 Simplify Log_{3}^{9 }+ Log_{3}^{21} – Log_{3}^{7}
Solution:
Log_{3}^{9 }+ Log_{3}^{21} – Log_{3}^{7}
= Log_{3}(9×21÷7)
= Log_{3}(9×21/7)
= Log_{3}27
= Log_{3}3^{3} = 3 Log _{3}3
But Log_{3}3 = 1
Therefore 3 x 1 =3
Log_{3}^{9 }+ Log_{3}^{21} – Log_{3}^{7 }= 3
 Solve completely for x in the equation 4Log_{x}5 = Log_{5}x
Solution:
4Log_{x}5 = Log_{5}x
4Log_{x}5 = 4/Log_{5}x since Log_{y}x = 1/Log_{x}y
therefore 4/Log_{5}x = Log_{5}x = 4 = (Log_{5}x)2
take square root of both sides
x = ±2
Therefore Log_{5}x = ±2
Hence, Log_{5}x = 2 or Log_{5}x = 2
x = 5^{2} or 5^{2}
x = 25 or 1/25
 Solve the equation Log_{4(}x^{2} + 6x + 11) = 1/2
Solution:
Log_{4(}x^{2} + 6x + 11) = 1/2
x^{2} + 6x + 11 = 4^{1/2} = 2
x^{2} + 6x + 11 – 2 = 0
x^{2} + 6x + 9= 0
(x +3)(x + 3) = 0
x = 3 twice
CLASS ACTIVITIES:
 Simplify the following
(a) Log_{3}27 + 2Log _{3}9 (b) Log _{x }x9 (c) Log_{5}
 Solve the following
(i) Log_{10}(x^{2}+4) = 2 + Log_{10}x – Log_{10}20 (ii) Log_{2}2n – 2Log_{8}n = 4
SUB TOPIC CHANGE OF BASE OF LOGARITHMS
To change the base of logarithms, we follow the procedure below
Let Log_{a}x= y then x = _{a}y
Log_{b}x = yLog_{b}a
yLog_{b}a = Log_{b}x
y = Log_{b}x/Log_{b}a
Therefore Log_{a}x = Log_{b}x/Log_{b}a
Example:
Change the base of the following logarithms to base 10
 Log_{3}81 (b) Log_{5}125 (c) Log_{c}x = d
Solution
 Log_{3}81 = x
Log_{3}81 = Log_{10}81/ Log_{10}3
= Log_{3}3^{4}/ Log_{10}3
= 4Log_{10}3/ Log_{10}3 = 3
Therefore Log_{3}81 = 4
 Log_{5}125 = Log_{3}125/ Log_{10}5
Log_{5}125 = Log_{10}5^{3}/ Log_{10}5
Log_{5}125 = 3Log5/ Log5
Therefore Log_{5}125 = 3
 Log_{c}x = d
d = Log_{10}x/ Log_{10}c
therefore Log_{c}x = Log_{10}x/ Log_{10}c
CLASS ACTIVITICES:
 Show that
SUB TOPIC: USE OF TABLES (GREATER THAN ONE AND LESS THAN ONE)
Generally logarithms with base 10 are universal. This is why we have the table of logarithms in base ten. The logarithm of any number has two parts. These are the characteristics and mantissa. The characteristic is the integer part.
Consider the Log_{10}530, Log_{10}53 and Log_{10}5.3.
The mantissas of all of them are the same. The difference is in the characteristics.
530 = 5.3 x 10^{2 }, 53 = 5.3 x 10^{1} whereas 5.3 = 5.3 x 10^{0}.
No Log
530 2.7243
53 1.7243
5.3 0.7243
This is useful to evaluate problems of multiplication and division
To check from table of logarithms, you will need to follow the examples
Example:
 use tables to evaluate the following
 65.43 x 1453 (b) 86.31 x 0.6218 (c) 0.07304 ÷ 0.8931
Solution:
No Standard form Log
65.43 6.543 x 10^{1} 1.815
1453 1.453 x 10^{3 }3.1623
4.9781
we look up 65 under difference 3
8 1 5 6 Add the difference
 2
8 1 5 8
14 under five difference 3
16.14 + 9 = 1623
–For multiplication we add
–Look up the antilog table
0.97 under 8 difference 1
9506 + 2 = 9508
The characteristic is used to determine the decimal point location.
Antilog of 0.9781 = 9.508, Hence 9.508 x 10^{4} = 95080
65.43 x 1453 = 95080
 86.31 x 0.6218
No Standard form Log
86.31 8.631 x 10^{1} 1.9361
0.6218 6.218 x 10^{1 }1.7937
1.7298
note: 1 is called bar 1.
Mantissa is always positive but characteristic can be negative or positive; we put the negative sign on it. When there is no sign then, it is positive.
9360 + 1 = 9361
7931 + 6 = 7937
From Antilog table
.7298 = 5358 + 10 = 5368

 x 10^{1 }= 53.6886.31 x 0.6218 = 53 .68
 0.07304 ÷ 0.8931
No Standard form Log
0.07304 7.304 x 10^{2} 2.8635
0.8931 8.931 x 10^{1 }1.9509
2.9126 subtract for division
8633 + 2 = 8635
9509 + 0 = 9509
Antilog of .9126 = 8166 + 11 = 8177
8.177 x 10^{2} = 0.08177
CLASS ACTIVITIES:
Use tables to solve the following
 23.82 x 142.8
 0.03167 x 102.8 x 0.325
 14.87 ÷ 2.314
 (12.31)^{2}
 (33.28) ÷ 4.689
SUB TOPIC: LOGARITHMIC EQUATIONS
EXAMPLE 1: Solve for m in the equation
Solution
Equating powers
4(2m+3) =
 4(2m + 3) = 5m+1
12 – 1 = 5m + 8m
13 = 13m
1 = m
EXAMPLE 2: Given that , solve for x and y respectively.
Solution
Applying the laws of logarithm
We can now equate terms
8 = yx – x ………………………………………………….. (i)
Also,
Applying the laws of logarithm,
We can now equate terms
4 = yx + x ………………………………………………………(ii)
Solving (i) and (ii) simultaneously, we eliminate yx.
8 = yx – x
(4 = yx + x)
4 = 2x
X = = – 2
Put x = – 2 into (2)
4 = yx + x will become
4 = y( 2) + ( 2)
4 = – 2y – 2
4 + 2 = – 2y
6 = – 2y
y =
x, y means – 2, – 3.
CLASS ACTIVITIES:
 Find n if
 If
PRACTICE EXERCISE
Objective test:
Choose the correct answer from the alternatives
 Evaluate Log_{0.25}8 (a) ½ (b)2/3 (c) 2/3 (d) 3/2
 If 2Log_{4}2 = x + 1, find the value of x. (a) 2 (b)1 (c) 0 (d) 1
 Given that Log_{3}(xy) = 1 and Log_{3}(2x +y) = 2, find the value of x. (a) 1 (b)2 (c) 3 (d) 4
Essay Questions:
Solve for x giving your answer correct to 3 S. F.
 2Log_{10}x + 3Log_{10}5 = 2
 Using logarithm table, evaluate
÷ correct to 3 s.f
 If , find x and y respectively.
 Given that
 Using logarithm table to evaluate
ASSIGNMENT
 Without using tables, simplify
 If m and n are positive real numbers such that
Find: (i) a relation between m and n which does not involve logarithms; (ii) the value of n if m = 16. (leave your answer as in surd in the simplest form).
 If
 Using logarithms table, evaluate giving your answer to three significant figures.
 Use tables to evaluate 13.81×142.8
KEY WORDS
 LOGARITHM
 ANTILOGARITHM
 CHARACTERICTICS
 MANTISSA
 DIFFERENCE
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