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The common logarithm is the logarithm to base 10. The notation 
is used by physicists, engineers, and calculator keypads to denote the common logarithm.
However, mathematicians generally use the same symbol to mean the natural
logarithm ln, 
. Worse still, in Russian literature the notation 
is used to denote a base-10 logarithm, which conflicts with
the use of the symbol lg to indicate the logarithm to base
2. To avoid all ambiguity, it is best to explicitly specify 
when the logarithm to base 10 is intended. In this
work, 
,
is used for the natural
logarithm, and 
is used for the logarithm to the base 2.
The situation is complicated even more by the fact that number theorists (e.g., Ivić 2003) commonly use the notation 
to denote the nested
natural logarithm 
.
The common logarithm is implemented in the Wolfram Language as Log[10, x] and Log10[x].
Hardy and Wright (1979, p. 8) assert that the common logarithm has "no mathematical interest."
Common and natural logarithms can be expressed in terms of each other as
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and
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The common logarithm extended into the complex plane is illustrated above.
." 5 Nov 2003.