Logarithmic Functions (Cambridge International A-Level Mathematics)

Logarithmic functions

What are logarithmic functions?

• $a^{x}=b$ and $\mathrm{log}_{a}b=x$ are equivalent statements

• a > 0

• a is called the base

• Every time you write a logarithm statement say to yourself what it means

  • log₃81 = 4 “the power you raise 3 to, to get 81, is 4”
  • log_pq = r “the power you raise p to, to get q, is r”

How are logarithms and powers inverses of each other?

• A logarithm is the inverse of raising to a power so we can use rules to simplify logarithmic functions

How do I use the inverse property of logs and powers?

• Recognising the rules of logarithms allows expressions to be simplified

• Recognition of common powers helps in simple cases For more awkward cases a calculator is needed

  • 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

• 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

What notation is used with logarithms?

• 10 is a common base

  • log₁₀x is abbreviated to log x or lg x

• The value e is another common base

  • log_ex is abbreviated to ln x

• (log x)²≠ log x²

Worked Example