Exponential Equations (Edexcel IAL Maths)

Spec point — 5.3

Exponential equations

What are exponential equations?

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

• See Exponential Functions

How do I solve exponential equations in the form a^f(x) = b?

• If the value of b can be written as a power of a (b = a^k )

  • Write $a^{f(x)}=a^{k}$
  • Solve $f(x)=k$

• If the value of b can not be written as a power of a

  • Apply logs of base a to both sides to get:

    • $f(x)=\mathrm{log}_{a}b$

  • Solve for x

How do I solve exponential equations in the form a^f(x) = b^g(x)?

• If the value of b can be written as a power of a (b = a^k )

  • Use the index law to rewrite $b^{g(x)}$

    • $b^{ \mathrm{g} ( x )} = ( a^{k )^{g} ( x )} = a^{k \mathrm{g} ( x ) }$

  • Write $a^{ \mathrm{f} ( x )} = a^{k \mathrm{g} ( x ) }$

  • Solve $f(x)=kg(x)$

• 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:

    • $\log ( a^{ \mathrm{f} ( x )} ) = \log ( b^{ \mathrm{g} ( x )} )$

  • Use the laws of logarithms to bring the power to the front:

    • $\mathrm{f} ( x ) \log a = \mathrm{g} ( x ) \log b$

  • log a and log b are just numbers so rearrange and solve for x

• If either side is multiplied by a constant ($pa^{f(x)}$ )

  • Do not write as $(pa)^{f(x)}$ – this is incorrect

  • Still take logs of both sides but you will need to use another law of logs

    • $\log ( p a^{ \mathrm{f} ( x )} ) = \log p + \log p a^{ \mathrm{f} ( x )}$

  • 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 $a^{ \mathrm{f} ( x )} + a^{ \mathrm{g} ( x )} + c = 0$

  • Use the index laws in reverse to split up the powers

    • $a^{Ax+B}=a^{Ax}\times a^{B}$

  • 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

Exam Hint

• 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