Exponential Equations | Edexcel International AS Maths: Pure 2 Revision Notes 2019

Exponential Equations Notes fig2, A Level & AS Maths: Pure revision notes

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:
  - $f (x) = \log_{a} b$
- Solve for x

If the value of b can be written as a power of a (b = a^k)
- Use the index law to rewrite $b^{g (x)}$
  - $b^{g (x)} {= (a^{k})}^{g (x)} = a^{k g (x)}$
- Write $a^{f (x)} = a^{k g (x)}$
- Solve $f (x) = k g (x)$
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:
  - $\log (a^{f (x)}) = \log (b^{g (x)})$
- Use the laws of logarithms to bring the power to the front:
  - $f (x) \log a = g (x) \log b$
- log a and log b are just numbers so rearrange and solve for x
If either side is multiplied by a constant ( $p a^{f (x)}$ )
- Do not write as ${(p a)}^{f (x)}$ – this is incorrect
- Still take logs of both sides but you will need to use another law of logs
  - $\log (p a^{f (x)}) = \log p + \log p a^{f (x)}$
- log p is just a constant so can be solved in the same way as above

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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

Exponential Equations (Edexcel International AS Maths: Pure 2)