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The logarithm of a positive real number to a positive base is the exponent for which . Therefore, for real and positive ,

and for positive and positive ,

The equation is called logarithmic form, while the equivalent equation is called exponential form (Abramson 2021). For example, and express the same relation. These forms reflect the inverse function relation between the logarithm and the exponential function.

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 literature.

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 .

In the Wolfram Language, the logarithm to the base is implemented as Log[b, x], while Log[x] gives the natural logarithm, i.e., Log[E, x], where E is the Wolfram Language symbol for e.

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

The logarithm can also be defined for complex arguments, as shown above. If the logarithm is taken as the forward function, the function taking the base to a given power is then called the antilogarithm.

For , is called the characteristic, and is called the mantissa.

Division and multiplication identities for the logarithm can be derived from the identity

Specifically, for ,

There are a number of properties which can be used to change from one logarithm base to another, including

An interesting property of logarithms follows from looking for a number such that

so

Another related identity that holds for arbitrary is given by

Numbers of the form are irrational if and are integers, one of which has a prime factor which the other lacks. A. Baker made a major step forward in transcendental number theory by proving the transcendence of sums of numbers of the form for and algebraic numbers.

See also

Antilogarithm, Base, Cologarithm, Common Logarithm, e, Exponential Form, Exponential Function, Harmonic Logarithm, Lg, Ln, Logarithmic Form, Logarithmic Number, Logarithm Laws, Logarithmic Series, Napierian Logarithm, Natural Logarithm, Nested Logarithm, Power Explore this topic in the MathWorld classroom

Related Wolfram sites

https://functions.wolfram.com/ElementaryFunctions/Log/

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Logarithmic Function." §4.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 67-69, 1972.Abramson, J. "Logarithmic Functions." §6.3 in Algebra and Trigonometry, 2nd ed. Houston, TX: OpenStax, 2021. https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-3-logarithmic-functions.Beyer, W. H. "Logarithms." CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 159-160 and 221, 1987.Conway, J. H. and Guy, R. K. "Logarithms." The Book of Numbers. New York: Springer-Verlag, pp. 248-252, 1996.Ivić, A. "On a Problem of Erdős Involving the Largest Prime Factor of ." 5 Nov 2003. https://arxiv.org/abs/math/0311056.Numberphile. "Log Tables." Sep. 5, 2013. https://www.youtube.com/watch?v=VRzH4xB0GdM.Pappas, T. "Earthquakes and Logarithms." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 20-21, 1989.Spanier, J. and Oldham, K. B. "The Logarithmic Function ." Ch. 25 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 225-232, 1987.

Referenced on Wolfram|Alpha

Logarithm

Cite this as:

Weisstein, Eric W. "Logarithm." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Logarithm.html

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