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
  • log_e?x?is abbreviated to?ln x
• (log x)²?≠ log x²