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Exponential Equations (Cambridge (CIE) O Level Additional Maths): Revision Note

Exam code: 4037

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Solving exponential equations

What are exponential equations?

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

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

    • For example,

table row cell 5 to the power of 2 x end exponent end cell equals 125 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

  • In more complicated cases the laws of logarithms should be used to solve exponential equations

  • The change of base law can be used to solve some exponential equations without a calculator

    • For example,

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

How do we use logarithms to solve exponential equations?

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

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

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

    • ln (loge) is often used

    • The answer is often written in terms of ln

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

  • Follow these steps to solve exponential equations

    • STEP 1: Take logarithms of both sides

    • STEP 2: Use the laws of logarithms to remove the powers

    • STEP 3: Rearrange to isolate x

    • STEP 4: Use logarithms to solve for x

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

      • 4x = (22)x = 22x = (2x)2

      • e 2x = (e2)x = (ex)2

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 by noticing that 4 to the power of x space equals space open parentheses 2 squared close parentheses to the power of x space equals space open parentheses 2 to the power of x close parentheses squared.

Rewrite the first term as a power of 2.

open parentheses 2 to the power of x close parentheses squared space minus space 3 open parentheses 2 to the power of x space plus space 1 end exponent close parentheses space plus space 9 space equals space 0

Rewrite the middle terms using the laws of indices: If  2 to the power of x plus 1 end exponent space equals space 2 to the power of x space cross times space 2 to the power of 1 space equals space 2 open parentheses 2 to the power of x close parentheses     

 open parentheses 2 to the power of x close parentheses squared space minus space 3 cross times 2 open parentheses 2 to the power of x close parentheses space plus space 9 space equals space 0 open parentheses 2 to the power of x close parentheses squared space minus space 6 open parentheses 2 to the power of x close parentheses space plus space 9 space equals space 0    

USing a substitution can make this easier to solve. 

Let u space equals space 2 to the power of x

open parentheses u close parentheses squared space minus space 6 open parentheses u close parentheses space plus space 9 space equals space 0

Factorise.

u squared space minus space 6 u space plus space 9 space equals space 0 open parentheses u space minus space 3 close parentheses open parentheses u space minus space 3 close parentheses space equals space 0

Solve to find u and substitute 2x back in.

table row cell u space end cell equals cell space 3 end cell row cell 2 to the power of x space end exponent end cell equals cell space 3 end cell end table

Solve the exponential equation 2x = 3 by taking logarithms of both sides.

ln space 2 to the power of x space equals space ln space 3

Bring the power down using the law of logs ln space x to the power of m space equals space m ln space x.

x ln space 2 space equals space ln space 3

Rearrange and solve. 

x space equals space fraction numerator ln space 3 over denominator ln space 2 end fraction equals space 1.584...

bold italic x bold space bold equals bold space bold 1 bold. bold 58 bold space begin bold style stretchy left parenthesis 3 space s. f. stretchy right parenthesis end style

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    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.