SS1 First Term Further Mathematics Lesson Note – Logarithms

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WEEK TWO

LOGARITHMS

CONTENTS:

1. Laws of Logarithms
2. Change of Base of Logarithms.
3. Use of Tables (greater than one and less than one).
4. Logarithmic Equations.

SUB TOPIC: LAWS OF LOGARITHMS

In the last topic indices, we learnt that p = a^x , e. g 1000 = 10³ 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.

Log₂8 = 3 because 8 = 2³

Log ₃9 = 2 because 9 = 3²

 

Laws of Logarithms

1. Log_a(pq) = Log_ap + Log_aq = Multiplication rule

e.g. if Log₃(6×5) = Log₃6 + Log₃5

2. Log_aa = 1 e.g Log₁₀10 = 1
3. Log_a(x/y) = Log_ax – Log_ay = Division Rule
4. Log_a(x)ⁿ = nLog_ax
5. Log_a1 = 0
6. Log_a(1/x) = Log_ax^-1 = -1Log_ax
7. If Log_by = 1 then y = b
8. If Log_ba = 1/Log_ab
9. Log_b = 1/nLog_bx
10. Log_b Log_yb = 1 for b and y positive and not equal to 1
11. Log _by = Log _by / Log _a

Examples:

1. Simplify Log₃⁹ + Log₃²¹ – Log₃⁷

Solution:

Log₃⁹ + Log₃²¹ – Log₃⁷

= Log₃(9×21÷7)

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