2.1.2 Logarithmic Functions

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

• As a logarithm is the inverse of raising to a power
  
• There are more laws of logarithms

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