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

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What are exponential equations? 

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

 Exponential Equations Notes fig2, A Level & AS Maths: Pure revision notes 

  • See Exponential Functions

How do I solve exponential equations of the form af(x) = b?

  • If the value of b can be written as a power of a (ak )
    • Write a to the power of straight f left parenthesis x right parenthesis end exponent equals a to the power of k
    • Solve straight f left parenthesis x right parenthesis equals k
  • If the value of b can not be written as a power of a
    • Apply logs of base a to both sides to get:
      • straight f left parenthesis x right parenthesis equals log subscript a b
    • Solve for x

How do I solve exponential equations of the form af(x) = bg(x)?

  • If the value of b can be written as a power of a (ak )
    • Use the index law to rewrite b to the power of straight g left parenthesis x right parenthesis end exponent
      • b to the power of straight g left parenthesis x right parenthesis end exponent equals open parentheses a to the power of k close parentheses to the power of straight g left parenthesis x right parenthesis end exponent equals a to the power of k straight g open parentheses x close parentheses blank end exponent
    • Write a to the power of straight f open parentheses x close parentheses end exponent equals a to the power of k straight g open parentheses x close parentheses blank end exponent
    • Solve straight f stretchy left parenthesis x stretchy right parenthesis equals k straight g left parenthesis x right parenthesis
  • 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:
      • log invisible function application open parentheses a to the power of straight f open parentheses x close parentheses end exponent close parentheses equals log invisible function application open parentheses b to the power of straight g open parentheses x close parentheses end exponent close parentheses
    • Use the laws of logarithms to bring the power to the front:
      • straight f open parentheses x close parentheses log invisible function application a equals straight g open parentheses x close parentheses log invisible function application b
    • log a and log b are just numbers so rearrange and solve for x
  • If either side is multiplied by a constant (space p a to the power of straight f left parenthesis x right parenthesis end exponent )
    • Do not write as left parenthesis p a right parenthesis to the power of straight f left parenthesis x right parenthesis end exponent – this is incorrect
    • Still take logs of both sides but you will need to use another law of logs
      • log invisible function application open parentheses p a to the power of straight f open parentheses x close parentheses end exponent close parentheses equals log invisible function application p plus log invisible function application p a to the power of straight f open parentheses x close parentheses end exponent
    • 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 a to the power of straight f open parentheses x close parentheses end exponent plus a to the power of straight g open parentheses x close parentheses end exponent plus c equals 0
    • Use the index laws in reverse to split up the powers
      • a to the power of A x plus B end exponent equals a to the power of A x end exponent cross times a to the power of B
    • 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

Worked example

5-1-4-exp-equations-we-solution

Examiner Tip

  • Pay attention to how the question asks you to write your answer
    • it could ask for exact form, a specific form or rounded to a specified degree of accuracy

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Paul

Author: Paul

Expertise: Maths

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.