Exponentials and Logarithms
An exponential function is one a constant is raised to the power of a variable:
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:
The gradient graphs of an exponential function are always very similar, but when a = 2.72 (e), the gradient graph is exactly the same:
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 are the inverse of exponential functions.
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: