Editing Discrete logarithm (section)

=== Powers of 10 ===
The [[power of 10|powers of 10]] are
:<math>\ldots, 0.001, 0.01, 0.1, 1, 10, 100, 1000, \ldots.</math>
For any number <math>a</math> in this list, one can compute <math>\log_{10}a</math>. For example, <math>\log_{10}{10000}=4</math>, and <math>\log_{10}{0.001}=-3</math>. These are instances of the discrete logarithm problem.

Other base-10 logarithms in the real numbers are not instances of the discrete logarithm problem, because they involve non-integer exponents. For example, the equation <math>\log_{10}{53}=1.724276\ldots</math> means that <math>10^{1.724276\ldots}</math>. While integer exponents can be defined in any group using products and inverses, arbitrary real exponents, such as this 1.724276…, require other concepts such as the [[exponential function]].

In [[group-theoretic]] terms, the powers of 10 form a [[cyclic group]] <math>G</math> under multiplication, and 10 is a [[cyclic group|generator]] for this group. The discrete logarithm <math>\log_{10}a</math> is defined for any <math>a</math> in <math>G</math>.