The Number e (AQA Level 3 Mathematical Studies (Core Maths)): Revision Note

The Number e

What is e, the exponential function?

• The exponential function is

  • is an irrational number

  • is approximately 2.718...

• As with other exponential graphs

  • passes through (0, 1)

  • has the -axis as an asymptote

What makes e special?

• has the particular property that:

  • The gradient at any point on the graph is equal to the -value at that point

  • I.e. The gradient of is at all real values of

  • It is also true that the gradient of is where is a constant

• There are many natural phenomenon which demonstrate this relationship, known as exponential growth

  • E.g. The growth of bacteria or the populations of species

• Consider the following table of values for

The negative exponential graph

• is a reflection in the -axis of 

• There are many natural phenomenon which demonstrate this relationship, known as exponential decay

  • E.g. The decay of radioactive substances

• The gradient of is

  • I.e. The gradient is equal to the -value, but negative, as it is a downward slope

    

Solving Equations Involving e

What is a natural logarithm?

• The same concept as outlined in Exponential Functions & Logarithms can be used

• Use the relationship that

  • If  then 

• This can be used when the base is

  • If then

• When working with , we can use "natural log" to denote log to base

  • If then

  • There should be a button on your calculator which looks similar to

How do I solve an equation involving e?

• Use the relationship:

  • If  then 

  • To find any unknowns

• For example

  • can be rewritten as a natural logarithm

    • 

    • This can then be entered into your calculator using the button

    • so to 3 significant figures

  • can also be rewritten as a natural logarithm

    • 

    • This can then be rearranged for

    • 

    • 

    • So to 3 significant figures