# AQA A Level Maths: Pure复习笔记6.1.2 Logarithmic Functions

### 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”

#### Logarithm rules

• A logarithm is the inverse of raising to a power so we can use rules to simplify logarithmic functions • There are more laws of logarithms but these are the only ones needed for A level

#### 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

For more awkward cases a calculator is needed • 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  