IBMaths: AA HLDPRevision Notes1. Number & Algebra1.2 Exponentials & Logs1.2.3 Solving Exponential Equations

Solving Exponential Equations (DP IB Maths: AA HL)

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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 attributes columnalign right center left columnspacing 0px end attributes 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 attributes columnalign right center left columnspacing 0px end attributes 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

Exam Tip

  • 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.

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

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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.