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

Logarithmic Functions

Logarithmic functions

6.1.2-Logarithmic-Functions-Notes-fig1

  • 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 functions6.1.2-Logarithmic-Functions-Notes-fig2
  • There are more laws of logarithms but these are the only ones needed for A level

How do I use logarithms?

6.1.2-Logarithmic-Functions-Notes-fig3

  •  Recognising the rules of logarithms allows expressions to be simplified

6.1.2-Logarithmic-Functions-Notes-fig4

  • 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 6.1.2-Logarithmic-Functions-Notes-fig5

  • Calculators can have, possibly, three different logarithm buttons

6.1.2-Logarithmic-Functions-Notes-fig6

  • This button allows you to type in any number for the base

6.1.2-Logarithmic-Functions-Notes-fig7

  • Natural logarithms (see “e”)

6.1.2-Logarithmic-Functions-Notes-fig8

  • Shortcut for base 10 although SHIFT button needed
  • Before calculators, logarithmic values had to be looked up in printed tables

Notation

6.1.2-Logarithmic-Functions-Notes-fig9

  • 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

6.1.2-Logarithmic-Functions-Example-fig1

 

 

 

 

 

 

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