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

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"e"

What is e, the exponential function?

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

What is the big deal with e?

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)

The negative exponential graph

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)

Exponential growth and decay

y = Ae^kx (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

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

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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"
- $\ln x \equiv \log_{e} x$
- 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
- $\ln 1 = 0$
- $\ln e = 1$
- $\ln e^{x} = x$
- $\ln x$ 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
- $\log_{a} b = \frac{\ln b}{\ln a}$

How can I solve equations involving e & ln?

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 $f (x) = e^{k}$

Worked example

3-1-1-ln-we-solution

Examiner Tip

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 $\frac{1}{2} \ln 25$

e & ln (Edexcel International A Level Maths: Pure 3)

Revision Note

What is e, the exponential function?

What is the big deal with e?

The negative exponential graph

Exponential growth and decay

What is ln?

What are the properties of ln?

How can I solve equations involving e & ln?

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1. Algebra & Functions

1.1 Rational Expressions

1.1.1 Rational Expressions

1.1.2 Top Heavy Rational Expressions

1.2 Functions

1.2.1 Language of Functions

1.2.2 Composite Functions

1.2.3 Inverse Functions

1.2.4 Modulus Functions - Sketching Graphs

1.2.5 Modulus Functions - Solving Equations

1.3 Combinations of Transformations

1.3.1 Combinations of Transformations

2. Trigonometry

2.1 Reciprocal & Inverse Trigonometric Functions

2.1.1 Reciprocal Trig Functions - Definitions

2.1.2 Reciprocal Trig Functions - Graphs

2.1.3 Trigonometry - Further Identities

2.1.4 Inverse Trig Functions

2.2 Compound & Double Angle Formulae

2.2.1 Compound Angle Formulae

2.2.2 Double Angle Formulae

2.2.3 R addition formulae Rcos Rsin etc

2.2.4 Modelling with Trigonometric Functions

2.3 Further Trigonometric Equations

2.3.1 Strategy for Further Trigonometric Equations

2.4 Trigonometric Proof

2.4.1 Trigonometric Proof

3. Logs & Exponentials

3.1 Exponential & Logarithms

3.1.1 e & ln

3.1.2 Derivatives of Exponential Functions

3.2 Modelling with Exponentials & Logarithms

3.2.1 Exponential Growth & Decay

3.2.2 Using Exps & Logs in Modelling

3.2.3 Using Log Graphs in Modelling

4. Differentiation

4.1 Further Differentiation

4.1.1 Differentiating Other Functions (Trig, ln & e etc)

4.1.2 Chain Rule

4.1.3 Product Rule

4.1.4 Quotient Rule

4.1.5 Differentiating Reciprocal and Inverse Trig Functions

5. Integration

5.1 Further Integration

5.1.1 Integrating Other Functions (Trig, ln & e etc)

5.1.2 Reverse Chain Rule

5.1.3 f'(x)/f(x)

5.1.4 Integrating with Trigonometric Identities

6. Numerical Methods

6.1 Numerical Methods

6.1.1 Change of Sign

6.1.2 Change of Sign Failure

6.1.3 x = g(x) Iteration

Author: Dan