Exponential Equations (Edexcel International A Level Maths): Revision Note

Exponential Equations

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