5.1.2 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
  

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,  different logarithm buttons

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

• 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

• (log x)² ≠ log x²

What is ln? 

• ln is a function that stands for natural logarithm
• It is a logarithm where the base is the constant "e"
  • 
  • It is important to remember that ln is a function and not a number

What are the properties of ln? 

• Using the definition of a logarithm you can see
  • 
  • 
  •  
  • is only defined for positive x
• As ln is a logarithm you can use the laws of logarithms
  • 
  • 
  • 

How can I solve equations involving e & ln? 

• The functions and  are inverses of each other
  • If  then 
  • If  then 
• If your equation involves "e" then try to get all the "e" terms on one side
  • If "e" terms are multiplied, you can add the powers
    •  
    • You can then apply ln to both sides of the equation
  • If "e" terms are added, try transforming the equation with a substitution
    • For example: If  then
    • You can then solve the resulting equation (usually a quadratic)
    • Once you solve for y then solve for x using the substitution formula
• If your equation involves "ln", try to combine all "ln" terms together
  • Use the laws of logarithms to combine terms into a single term
  • If you have  then solve 
  • If you have  then solve