Functions, exponential and logarithm rules

Exponential rules

\[a^m \times a^n = a^{m + n}\]
\[a^{-m} = \frac{1}{a^m}\]
\[\frac{a^m}{a^n} = a^{m – n}\]
\[(a^m)^n = a^{mn}\]
\[(ab)^m = a^m b^m\]
\[a^0 = 1\]

Logarithm rules

\[log (ab) = log(a) + log(b)\]
\[log \left(\frac{a}{b}\right) = log(a) – log(b)\]
\[log (a^n) = n log(a)\]
\[log_a (a) = 1\]
\[log_a(1) = 0\]

We often use natural logarithms. This is sometimes written as \(log_e(x)\) or more often \(ln(x)\). Sometimes, \(log(x)\) is used to refer to the natural logarithm but where possible we will try to use the unambiguous notation.

Summation and product functions

The summation function

The summation notation is a convenient and concise way of writing sums. For two integers, \(a\) and \(b\), with \(a\leq b\), and a function \(f(.)\) the general notation is:

\[\sum_{i=a}^b f(i) = f(a) + f(a+1) + \cdots + f(b)\]

So, for example,

\[\sum_{i=2}^3 \frac{e^i \lambda^i}{i!} = \frac{e^2 \lambda^2}{2!} + \frac{e^3 \lambda^3}{3!}\]

An example that you are likely to encounter frequently is the sum of \(x_1\), \(x_2\), until \(x_n\),

\[\sum_{i=1}^n x_i = (x_1 + x_2 + x_3 + \cdots + x_n)\]

Two useful rules are:

\[\sum_{i=1}^n (a x_i) = a \sum_{i=1}^{n} x_i\]


\[\sum_{i=1}^n a = n a\]

The product function

Similarly, the product notation allows us to summarise long multiplications. The product of \(x_1\), \(x_2\), etc. until \(x_n\), for example, can be written

\[\prod_{i=1}^n x_i = (x_1 \times x_2 \times x_3 \times \cdots \times x_n)\]

A useful multiplication rule is:

\[\prod_{i=1}^n (a x_i) = a^n \prod_{i=1}^n x_i\]