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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

to mean

, 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

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