Editing List of logarithmic identities (section)

=== Summation/subtraction ===
The following summation/subtraction rule is especially useful in [[probability theory]] when one is dealing with a sum of log-probabilities:

{|
|<math>\log_b (a+c) = \log_b a + \log_b \left(1 + \frac{c}{a}\right)</math>
|because
|<math>\left(a + c \right) = a \times \left(1 + \frac{c}{a} \right)</math>
|-
|<math>\log_b (a-c) = \log_b a + \log_b \left(1 - \frac{c}{a}\right)</math>
|because
|<math>\left(a - c \right) = a \times \left(1 - \frac{c}{a} \right)</math>
|}

Note that the subtraction identity is not defined if <math>a=c</math>, since the logarithm of zero is not defined. Also note that, when programming, <math>a</math> and <math>c</math> may have to be switched on the right hand side of the equations if <math>c \gg a</math> to avoid losing the "1 +" due to rounding errors.  Many programming languages have a specific <code>log1p(x)</code> function that calculates <math>\log_e (1+x)</math> without underflow (when <math>x</math> is small).

More generally:
<math display="block">\log _b \sum_{i=0}^N a_i = \log_b a_0 + \log_b \left( 1+\sum_{i=1}^N \frac{a_i}{a_0} \right) = \log _b a_0 + \log_b \left( 1+\sum_{i=1}^N b^{\left( \log_b a_i - \log _b a_0 \right)} \right)</math>