e & ln (Edexcel International A Level Maths): Revision Note

"e"

What is e, the exponential function?

• 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?

• 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

• 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

"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 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