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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 = ax , 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. Loga(pq) = Logap + Logaq = Multiplication rule

e.g. if Log3(6×5) = Log36 + Log35

  1. Logaa = 1 e.g Log1010 = 1
  2. Loga(x/y) = Logax – Logay = Division Rule
  3. Loga(x)n = nLogax
  4. Loga1 = 0
  5. Loga(1/x) = Logax-1 = -1Logax
  6. If Logby = 1 then y = b
  7. If Logba = 1/Logab
  8. Logb = 1/nLogbx
  9. Logb Logyb = 1 for b and y positive and not equal to 1
  10. Log by = Log by / Log a

Examples:

  1. Simplify Log39 + Log321 – Log37

Solution:

Log39 + Log321 – Log37

= Log3(9×21÷7)

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