Exponential Equations (Edexcel International A Level Maths: Pure 2)

What are exponential equations? 

• An equation where the unknown is a power
• In simple cases the solutions can be “spotted”

  

• See Exponential Functions

How do I solve exponential equations of the form a^f(x) = b?

• If the value of b can be written as a power of a (b = a^k )
  • Write 
  • Solve 
• If the value of b can not be written as a power of a
  • Apply logs of base a to both sides to get:
    • 
  • Solve for x

How do I solve exponential equations of the form a^f(x) = b^g(x)?

• If the value of b can be written as a power of a (b = a^k )
  • Use the index law to rewrite 
    • 
  • Write 
  • Solve 
• If the value of b can not be written as a power of a
  • Apply logs of the same base (any base will work) to both sides to get:
    • 
  • Use the laws of logarithms to bring the power to the front:
    • 
  • log a and log b are just numbers so rearrange and solve for x
• If either side is multiplied by a constant ( )
  • Do not write as  – this is incorrect
  • Still take logs of both sides but you will need to use another law of logs
    • 
  • log p is just a constant so can be solved in the same way as above

How do I solve exponential equations with three terms? 

• If the equation has three terms such as 
  • Use the index laws in reverse to split up the powers
    • 
  • Try and use a substitution to transform the equation into a quadratic
    • See Further Solving Quadratics (Hidden Quadratics)
    • Look out for terms where one power is double another power
  • Solve the quadratic
  • For each solution find the corresponding value(s) of x