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e & ln (Edexcel International A Level Maths: Pure 3)
Revision Note
"e"
What is e, the exponential function?
- The exponential function is y = ex
- e is an irrational number
- e ≈ 2.718
- As other exponential graphs do, y = ex
- passes through (0, 1)
- has the x-axis as an asymptote
What is the big deal with e?
- y = ex has the particular property
- dy/dx = ex
- ie for every real number x, the gradient of y = ex is also equal to ex
(see Derivatives of Exponential Functions)
The negative exponential graph
- y = e-x is a reflection in the y-axis of y = ex
- They are of the form y = f(x) and y = f(-x)(see Transformations of Functions - Reflections)
Exponential growth and decay
- y = Aekx (k > 0) is exponential growth
- y = Ae-kx (k > 0) is exponential decay
- A is the initial value
- k is a (usually positive) constant
- “-“ is used in the equation making clear whether it is growth or decay
Worked example
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"ln"
What is ln?
- ln is a function that stands for natural logarithm
- It is a logarithm where the base is the constant "e"
- It is important to remember that ln is a function and not a number
What are the properties of ln?
- Using the definition of a logarithm you can see
- is only defined for positive x
- As ln is a logarithm you can use the laws of logarithms
- Any logarithm can be written in terms of the natural logarithm using the change of base formula
How can I solve equations involving e & ln?
- The functions and are inverses of each other
- If then
- If then
- If your equation involves "e" then try to get all the "e" terms on one side
- If "e" terms are multiplied, you can add the powers
- You can then apply ln to both sides of the equation
- If "e" terms are added, try transforming the equation with a substitution
- For example: If then
- You can then solve the resulting equation (usually a quadratic)
- Once you solve for y then solve for x using the substitution formula
- If "e" terms are multiplied, you can add the powers
- If your equation involves "ln", try to combine all "ln" terms together
- Use the laws of logarithms to combine terms into a single term
- If you have then solve
- If you have then solve
Worked example
Examiner Tip
- Always simplify your answer if you can
- for example,
- you wouldn't leave your final answer as so don't leave your final answer as
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