Exponentials and Logarithms

Exponentials

Exponential functions take the form . Their features include:

A -intercept of
All
The -axis as an asymptote.
If , then increases as increases, giving exponential growth.
If , then decreases as increases, giving exponential decay.

Modelling

Exponential functions are used to model situations where the rate of growth or decay is proportional to the current amount. This is because the derivative of is . A model with equation has:

Initial value () of
A rate of change of .

Logarithms

Logarithms are the inverse of exponentials. Below are rules of logarithms:

Converting:
Multiplication
Division:
Exponentiation:

Other required knowledge:

For any base, and
Logarithms in base are written as .
Logarithms in base 10 are written as .

A logarithmic function of the form has:

An -intercept of
The -axis as an asymptote.

Straightening graphs

Exponential

For a graph , taking on both sides gives:

Thus when plotting against , the gradient is and -intercept is .

Polynomial

For a graph , taking on both sides gives:

Thus when plotting against , the gradient is and the -intercept is .