CSEC O Level Math training in Logarithms

Aaron Walcott, TT


What Is a Logarithm?

A logarithm answers the question:

“To what power must the base be raised to get this number?”

If: 23 = 8


Then: log⁡2 8 = 3


Logs are simply exponents done backwards.

They allow us to:
• Break apart multiplication

• Simplify division
• Bring powers down in front

They are especially useful in algebra, calculus, growth/decay, and exam questions involving
unknown exponents.




Worked Example 1 (Product Law)

Simplify: log⁡2 (8𝑥)


Split using product rule: = log⁡2 8 + log⁡2 𝑥
= 3 + log⁡2 𝑥




Worked Example 2 (Power Law)

Simplify: log⁡3 (92 )


Apply power law: = 2log⁡3 9
=2×2=4




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, Aaron Walcott, TT


Logarithm Pinch Points

• Forgetting that logs turn multiplication into addition

• Losing minus signs in quotient rule

• Forgetting log of 1 = 0
• Forgetting log of base = 1

• Not recognising when change-of-base is required

• Treating log (a + b) as log a + log b (this is incorrect)




LAWS OF LOGARITHMS


Product Law log⁡𝑎 (𝑥𝑦) = log⁡𝑎 𝑥 + log⁡𝑎 𝑦




𝑥
Quotient Law log⁡𝑎 (𝑦) = log⁡𝑎 𝑥 − log⁡𝑎 𝑦




Power Law log⁡𝑎 (𝑥 𝑘 ) = 𝑘log⁡𝑎 𝑥




Log of 1 log⁡𝑎 1 = 0




Log of Base log⁡𝑎 𝑎 = 1




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