Logarithmic Functions (Cambridge O Level Additional Maths)

What are logarithmic functions?

• A logarithm is the inverse of raising to a power 
• If  then 
  • 
  • is called the base of the logarithm
• Try to get used to ‘reading’ logarithm statements to yourself
  •   would be read as “the power that you raise ... to, to get ..., is ”
  • So  would be read as “the power that you raise 5 to, to get 125, is 3”

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

Why use logarithms?

• Logarithms allow us to solve equations where the exponent is the unknown value
  • We can solve some of these by inspection
    • For example, for the equation 2^x = 8 we know that x must be 3
  • Logarithms allow use to solve more complicated problems
    • For example, the equation 2^x = 10 does not have a clear answer
    • Instead, we can use our calculator to find the value of

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

What notation might I see with logarithms?

• 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

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
• The natural logarithm () and the exponential function () are inverses of each other 
• It is defined for all positive numbers ()
  • cannot be defined for negative numbers or 

What are the properties of ln? 

• Using the definition of a logarithm you can see
  • 
  • 
  •  
  • is only defined for positive x

How can I solve equations involving e & ln? 

• The functions and  are inverses of each other
  • If  then 
  • If  then 
  • 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