IBMathsDPAnalysis & Approaches (AA)SLRevision Notes1. Number & AlgebraExponentials & LogsSolving Exponential Equations

Solving Exponential Equations (DP IB Analysis & Approaches (AA)): Revision Note

Amber

Written by: Amber

Reviewed by: Mark Curtis

Updated on

DP Analysis & Approaches (AA): SLDP Analysis & Approaches (AA): HL

Did this video help you?

Solving Exponential Equations

What are exponential equations?

  • An exponential equation is an equation where the power is the unknown

    • 5 to the power of x equals 8

    • 7 equals 2 to the power of x plus 3 end exponent

How do I solve exponential equations?

  • Some exponential equations can be solved by inspection (seeing the answer)

    • e.g. 3 to the power of x equals 9 is solved by x equals 2

  • Others can be solved by changing the base

    • e.g. 3 to the power of x plus 1 end exponent equals 1 over 9 to the power of x is solved by writing 9 as open parentheses 3 squared close parentheses:

table row cell 3 to the power of x plus 1 end exponent end cell equals cell 1 over open parentheses 3 squared close parentheses to the power of x end cell row cell 3 to the power of x plus 1 end exponent end cell equals cell 1 over 3 to the power of 2 x end exponent end cell row cell 3 to the power of x plus 1 end exponent end cell equals cell 3 to the power of negative 2 x end exponent end cell row cell x plus 1 end cell equals cell negative 2 x end cell row cell 3 x end cell equals cell negative 1 end cell row x equals cell negative 1 third end cell end table

  • But some exponential equations have no obvious answers, in which case use logarithms

    • e.g. 2 to the power of x equals 5 so x equals log subscript 2 5 equals 2.321928...

      • Recall if a to the power of x equals b then x equals log subscript a open parentheses b close parentheses

How do I solve quadratic exponential equations?

  • A quadratic exponential equation is open parentheses a to the power of x close parentheses squared plus b open parentheses a to the power of x close parentheses plus c equals 0

    • This is a hidden quadratic in a to the power of x

      • You can solve it with the substitution u equals a to the power of x

      • This gives two separate exponential equations to solve

  • Beware: the open parentheses a to the power of x close parentheses squared term may be disguised as a to the power of 2 x end exponent

    • Use the index law a to the power of 2 x end exponent equals open parentheses a to the power of x close parentheses squared to correct this:

      • straight e to the power of 2 x end exponent equals open parentheses straight e to the power of x close parentheses squared

      • 3 to the power of 2 x end exponent equals open parentheses 3 to the power of x close parentheses squared

      • 16 to the power of x equals 4 to the power of 2 x end exponent equals open parentheses 4 to the power of x close parentheses squared (a change of base is also needed here)

Examiner Tips and Tricks

If an exam question asks for exact solutions, you may have to leave your answer in the form of a logarithm.

Worked Example

Solve the equation 4 to the power of x minus 3 open parentheses 2 to the power of x plus 1 end exponent close parentheses plus blank 9 equals 0.  Give your answer correct to three significant figures.

aa-sl-1-2-3-solving-exp-equations-we-solution

You've read 1 of your 5 free revision notes this week

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

  • I would just like to say a massive thank you for putting together such a brilliant, easy to use website.I really think using this site helped me secure my top gradesin science and maths. You really did save my exams! Thank you.

    Beth
    IGCSE Student

  • This website is soooo useful and I can’t ever thank you enough for organising questions by topic like this. Furthermore, the name of the website could not have been more appropriate as it literally did SAVE MY EXAMS!

    Fathima
    A Level Student
Test yourselfFlashcards

Did this page help you?

Previous:Laws of LogarithmsNext:Language of Sequences & Series
Author
Reviewer
Amber

Author: Amber

Expertise: Maths Content Creator

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.

Mark Curtis

Reviewer: Mark Curtis

Expertise: Maths Content Creator

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.

Download notes on Solving Exponential Equations