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  • A-Level Maths

Exponentials and Logarithms

An exponential function is one a constant is raised to the power of a variable:

exponential functions, exponential graphs, e^x, e, A-Level Maths Notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes
  • The larger the coefficient, the steeper the graph

  • All exponential functions in the form y = a^x pass through (0, 1)

  • The value of the function decreases as x tends to 0

Functions in the form y = a^x where 0 < a < 1 are the other way around:

exponential functions, exponential graphs, e^x, e, A-Level Maths Notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes

The gradient graphs of an exponential function are always very similar, but when a = 2.72 (e), the gradient graph is exactly the same:

exponential functions, exponential graphs, e^x, e, A-Level Maths Notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes

For all real values of x:

if f(x) = e^x, f'(x) = e^x
if f(x) = e^kx, f'(x) = ke^kx

Functions in e can be used to model growth or decline where the rate of increase in number is proportional to the number.




Logarithms

Logarithms are the inverse of exponential functions.

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Just as indices laws apply to exponentials, there are a series of laws that apply to logarithms - these are known as the log rules or laws of logarithms: