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Powers of 10

The powers of 10 are

For any number a in this list, one can compute log a. For example, log 10000 = 4, and log 0.001 = −3. 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 log 53 = 1.724276… means that 10 = 53. 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 G under multiplication, and 10 is a generator for this group. The discrete logarithm log a is defined for any a in G.