Laws of Logarithms (Edexcel A Level Maths: Pure)

What are the laws of logarithms?

• There are many laws or rules of indices, for example
  • a^m x aⁿ = a^m+n
  • (a^m)ⁿ = a^mn
• There are equivalent laws of logarithms (for a > 0)
  • 
  • 
  •  

• There are also some particular results these lead to
  • 
  • 
  • 
  • 
  • 

 

• Two of these were seen in the notes Logarithmic Functions
• Beware …
  • … log (x + y) ≠ log x + log y
• Results apply to ln too
  • 
  • In particular  and 

How do I use the laws of logarithms?

• Laws of logarithms can be used to …
  • … simplify expressions
  • … solve logarithmic equations
  • … solve exponential equations

   

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