For any base. the logarithm function has a singularity at
. In the above plot, the blue curve is the logarithm to base 2 (
), the black curve is the logarithm to base
(the natural logarithm
), and the red curve is the logarithm to base 10 (the common logarithm. i.e.
Note that while logarithm base 10 is denoted
in this work, on calculators, and in elementary algebra and calculus textbooks, mathematicians and advanced mathematics texts uniformly use the notation
, and therefore use
to mean the common logarithm. Extreme care is therefore needed when consulting the
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
Whereas powers of trigonometric functions are denoted using notations like
is less commonly used in favor of the notation
Logarithms are used in many areas of science and engineering in which quantities vary over a large range. For example, the decibel scale for the loudness of sound, the Richter scale of earthquake magnitudes, and the astronomical scale of stellar brightnesses are all logarithmic scales.
The derivative and indefinite integral of
are given by