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Exponential Equations (Edexcel International A Level Maths: Pure 2)
Revision Note
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 of the form af(x) = b?
- If the value of b can be written as a power of a (b = ak )
- Write
- Solve
- If the value of b can not be written as a power of a
- Apply logs of base a to both sides to get:
- Solve for x
- Apply logs of base a to both sides to get:
How do I solve exponential equations of the form af(x) = bg(x)?
- If the value of b can be written as a power of a (b = ak )
- Use the index law to rewrite
- Write
- Solve
- Use the index law to rewrite
- 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:
- Use the laws of logarithms to bring the power to the front:
- log a and log b are just numbers so rearrange and solve for x
- Apply logs of the same base (any base will work) to both sides to get:
- If either side is multiplied by a constant ( )
- Do not write as – this is incorrect
- Still take logs of both sides but you will need to use another law of logs
- 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
- Use the index laws in reverse to split up the powers
- 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
- Use the index laws in reverse to split up the powers
Worked example
Examiner Tip
- 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
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