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Solving Exponential Equations (Edexcel IGCSE Further Pure Maths)

Revision Note

Amber

Written by: Amber

Reviewed by: Dan Finlay

Solving Exponential Equations

What are exponential equations?

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

  • In simple cases the solution can be spotted without the use of a calculator

    • For example,

table row cell 3 to the power of 2 x end exponent end cell equals 27 row cell 3 cubed end cell equals cell 27 space space space so end cell row cell 2 x space end cell equals cell space 3 end cell row cell x space end cell equals cell space 3 over 2 end cell end table

  • The change of base law can also be used to solve some exponential equations

    • For example,

table row cell 27 to the power of x space end cell equals cell space 9 blank end cell end table

  • Rewrite using the definition of a logarithm

table row x equals cell log subscript 27 9 end cell end table

  • Then use the change of base formula from the exam formula sheet

    • Use base 3 here because 9 and 27 are both powers of 3

table row x equals cell blank fraction numerator log subscript 3 9 over denominator log subscript 3 27 end fraction end cell row blank equals cell 2 over 3 end cell end table

  • In more complicated cases use the laws of logarithms to solve exponential equations

How do I use logarithms to solve exponential equations?

  • An exponential equation can be solved by taking logarithms of both sides

    • ln, or log subscript straight e, is often used

      • Though a log to any base could be used

    • The laws of indices may be needed to rewrite the equation first

    • The laws of logarithms can then be used to solve the equation

    • A question may ask you to give your answer in a particular form

      • For example as an exact value in terms of ln

  • STEP 1
    Take logarithms of both sides

table attributes columnalign right center left columnspacing 0px end attributes row cell 5 to the power of x end cell equals 27 row cell ln open parentheses 5 to the power of x close parentheses end cell equals cell ln 27 end cell end table

  • STEP 2
    Use the laws of logarithms to move powers out of the logarithms

space x ln 5 equals ln 27

  • STEP 3
    Rearrange to isolate x

x equals fraction numerator ln 27 over denominator ln 5 end fraction

  • Note that this is the exact solution to the equation

  • STEP 4
    Use logarithms in your calculator to find the value of x

table attributes columnalign right center left columnspacing 0px end attributes row x equals cell 2.047818... end cell row blank equals cell 2.05 space open parentheses 3 space straight s. straight f. close parentheses end cell end table

  • Only perform this step if required by the question

What about hidden quadratics?

  • Look for 'hidden' squared terms that could be changed to form a quadratic

    • In particular look out for terms such as

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

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

    • This can be used to factorise quadratic expressions

      • 4 to the power of x minus 2 to the power of x equals open parentheses 2 to the power of x close parentheses squared minus 2 to the power of x equals 2 to the power of x open parentheses 2 to the power of x minus 1 close parentheses

      • straight e to the power of 2 x end exponent minus 4 straight e to the power of x minus 5 equals open parentheses straight e to the power of x close parentheses squared minus 4 straight e to the power of x minus 5 equals open parentheses straight e to the power of x plus 1 close parentheses open parentheses straight e to the power of x minus 5 close parentheses

Examiner Tips and Tricks

  • Always check which form the question asks you to give your answer in

    • This can help you decide how to solve it

  • If the question requires an exact value you may need to leave your answer as 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.

Spot the hidden quadratic:  4 to the power of x equals open parentheses 2 squared close parentheses to the power of x equals 2 to the power of 2 x end exponent equals open parentheses 2 to the power of x close parentheses squared

Also note that  2 to the power of x plus 1 end exponent equals 2 to the power of 1 cross times 2 to the power of x equals 2 cross times 2 to the power of x

Substitute these into the equation and simplify

table row cell open parentheses 2 to the power of x close parentheses squared minus 3 open parentheses 2 cross times 2 to the power of x close parentheses plus 9 end cell equals 0 row cell open parentheses 2 to the power of x close parentheses squared minus 6 open parentheses 2 to the power of x close parentheses plus 9 end cell equals 0 end table

Factorise the quadratic

The left-hand side is in the form  y squared minus 6 y plus 9  where y equals 2 to the power of x 
This factorises to open parentheses y minus 3 close parentheses squared

open parentheses 2 to the power of x minus 3 close parentheses squared equals 0

Solve for 2 to the power of x

table row cell 2 to the power of x minus 3 end cell equals 0 row cell 2 to the power of x end cell equals 3 end table

Take logarithms of both sides

ln open parentheses 2 to the power of x close parentheses equals ln 3

Use  log subscript a x to the power of k equals blank k log subscript a x  to take the power out of the logarithm

Remember that  ln space equals space log subscript straight e

space x ln 2 equals ln 3

Solve for x

x equals fraction numerator ln 3 over denominator ln 2 end fraction

Use your calculator to find the decimal equivalent of that exact answer

x equals 1.584962...

Round to 3 significant figures

bold italic x bold equals bold 1 bold. bold 58 (3 s.f.)

Once 2 to the power of x equals 3 is found, the logarithm x equals log subscript 2 open parentheses 3 close parentheses could be used instead to find the value of x 

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    Previous:Properties of LogarithmsNext:Simplifying Surds
    Author
    Reviewer
    Amber

    Author: Amber

    Expertise: Maths

    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.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.