Logarithmic functions

• a = b^x?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”
  • log_p?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
  • log_10?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