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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:

• 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

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: