3.1.1 e & ln | Edexcel International A Level Maths: Pure 3 Revision Notes 2020

"e"

e Notes fig1, A Level & AS Maths: Pure revision notes

The exponential function is y = e^x
- e is an irrational number
- e ≈ 2.718
As other exponential graphs do, y = e^x
- passes through (0, 1)
- has the x-axis as an asymptote

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

e Notes fig3, A Level & AS Maths: Pure revision notes

y = e^x has the particular property
- dy/dx = e^x
- ie for every real number x, the gradient of y = e^x is also equal to e^x
  (see Derivatives of Exponential Functions)

e Notes fig4, A Level & AS Maths: Pure revision notes

y = e^-x is a reflection in the y-axis of y = e^x
- They are of the form y = f(x) and y = f(-x)
  (see Transformations of Functions - Reflections)

e Example fig1, A Level & AS Maths: Pure revision notes e Example fig2, A Level & AS Maths: Pure revision notes

ln is a function that stands for natural logarithm
It is a logarithm where the base is the constant "e"
- $\ln x \equiv \log_{e} x$
- It is important to remember that ln is a function and not a number

Using the definition of a logarithm you can see
- $\ln 1 = 0$
- $\ln e = 1$
- $\ln e^{x} = x$
- $\ln x$ is only defined for positive x
As ln is a logarithm you can use the laws of logarithms
- $\ln a + \ln b = \ln (a b)$
- $\ln a - \ln b = \ln (\frac{a}{b})$
- $n \ln a = \ln (a^{n})$
Any logarithm can be written in terms of the natural logarithm using the change of base formula
- $\log_{a} b = \frac{\ln b}{\ln a}$

The functions $e^{x}$ and $\ln x$ are inverses of each other
- If $e^{f (x)} = g (x)$ then $f (x) = \ln g (x)$
- If $\ln f (x) = g (x)$ then $f (x) = e^{g (x)}$
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
  - $e^{x} \times e^{y} = e^{x + y}$
  - 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 $y = e^{x}$ then $e^{4 x} = y^{4}$
  - You can then solve the resulting equation (usually a quadratic)
  - Once you solve for y then solve for x using the substitution formula
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 $\ln f (x) = \ln g (x)$ then solve $f (x) = g (x)$
- If you have $\ln f (x) = k$ then solve $f (x) = e^{k}$

3-1-1-ln-we-solution

Always simplify your answer if you can
- for example, $\frac{1}{2} \ln 25 = \ln \sqrt{25} = \ln 5$
- you wouldn't leave your final answer as $\sqrt{25}$ so don't leave your final answer as $\frac{1}{2} \ln 25$