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Logarithmic Functions (CIE A Level Maths: Pure 3)
Revision Note
Logarithmic Functions
Logarithmic functions
- a = bx and log b a = x are equivalent statements
- a > 0
- b is called the base
- Every time you write a logarithm statement say to yourself what it means
- log3 81 = 4
“the power you raise 3 to, to get 81, is 4”
- logp q = r
“the power you raise p to, to get q, is r”
- log3 81 = 4
Logarithm rules
- A logarithm is the inverse of raising to a power so we can use rules to simplify logarithmic functions
How do I use logarithms?
- Recognising the rules of logarithms allows expressions to be simplified
- Recognition of common powers helps in simple cases
- Powers of 2: 20 = 1, 21 = 2, 22 = 4, 23 = 8, 24 =16, …
- Powers of 3: 30 = 1, 31 = 3, 32 = 9, 33 = 27, 34 = 81, …
- The first few powers of 4, 5 and 10 should also be familiar
- Calculators can have, possibly, three different logarithm buttons
- This button allows you to type in any number for the base
- Natural logarithms (see “e”)
- Shortcut for base 10 although SHIFT button needed
- Before calculators, logarithmic values had to be looked up in printed tables
Notation
- 10 is a common base
- log10 x is abbreviated to log x or lg x
- The value e is another common base
- loge x is abbreviated to ln x
- (log x)2 ≠ log x2
Worked example
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