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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.

logarithms, log rules, logarithm rules, A-level Maths Notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes

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:

logarithms, log rules, logarithm rules, A-level Maths Notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes

When working with difficult equations with exponents, it can be helpful to apply logarithms to both sides:

logarithms, log rules, logarithm rules, A-level Maths Notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes


The Natural Logarithm, ln

The natural logarithm, ln, is the logarithm in base e. The graph of ln(x) is reflection of e^x in the line y=x:

the natural logarithm, ln(x) graph, lnx graph, y=lnx, y = ln x, e vs ln. A-level Maths Notes, EngineeringNotes.net, EngineeringNotes, Engineering Notes

The y=axis is an asymptote in this graph, meaning that

ln(x) is only defined when x > 0


Logarithms and Non-Linear Data

Often, it is helpful to model non-linear data as linear. This can be achieved by taking logs of one or both sides of the equation:

Non linear data and logarithms, A-Level Maths Notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes
  • If the equation is in the form y = ax^n, a graph of log y against log x will give a straight line where log a is the y intercept and n the gradient.

Non linear data and logarithms, A-Level Maths Notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes
  • If the equation is in the form y = ab^x, a graph of log y against x will give a straight line where log a is the y intercept and log b the gradient.