Edexcel A Level Maths: Pure:复习笔记6.1.2 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