info prev up next book cdrom email home



The logarithm is defined to be the Inverse Function of taking a number to a given Power. Therefore, for any $x$ and $b$,

x=b^{\log_b x},
\end{displaymath} (1)

or equivalently,
\end{displaymath} (2)

Here, the Power $b$ is known as the Base of the logarithm. For any Base, the logarithm function has a Singularity at $x=0$. In the above plot, the solid curve is the logarithm to Base $e$ (the Natural Logarithm), and the dotted curve is the logarithm to Base 10 (Log).

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.

\begin{figure}\begin{center}\BoxedEPSF{LogReIm.epsf scaled 700}\end{center}\end{figure}

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 $x=\log N$, $\left\lfloor{x}\right\rfloor $ is called the Characteristic and $x-\left\lfloor{x}\right\rfloor $ is called the Mantissa. Division and multiplication identities follow from these

xy = b^{\log_b x}b^{\log_b y} = b^{\log_b x+\log_b y},
\end{displaymath} (3)

from which it follows that
\log_b(xy) = \log_b x+\log_b y
\end{displaymath} (4)

\log_b\left({x\over y}\right)= \log_b x-\log_b y
\end{displaymath} (5)

\log_b x^n = n \log_b x.
\end{displaymath} (6)

There are a number of properties which can be used to change from one Base to another
a = a^{\log_a b/\log_a b} = (a^{\log_a b})^{1/\log_a b} = b^{1/\log_a b}
\end{displaymath} (7)

\log_b a = { 1\over \log_a b}
\end{displaymath} (8)

\log_x z = \log_x(y^{\log_y z}) = \log_y z\log_x y
\end{displaymath} (9)

\log_y z = {\log_x z\over \log_x y}
\end{displaymath} (10)

a^x = b^{x/\log_a b} = b^{x \log_b a}.
\end{displaymath} (11)

The logarithm Base e is called the Natural Logarithm and is denoted $\ln x$ (Ln). The logarithm Base 10 is denoted $\log x$ (Log), (although mathematics texts often use $\log x$ to mean $\ln x$). The logarithm Base 2 is denoted $\lg x$ (Lg).

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

\end{displaymath} (12)

x+y={1\over x-y}
\end{displaymath} (13)

\end{displaymath} (14)

\end{displaymath} (15)

\log_b(x+\sqrt{x^2-1}\,) = -\log_b(x-\sqrt{x^2-1}\,).
\end{displaymath} (16)

Numbers of the form $\log_a b$ are Irrational if $a$ and $b$ 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 $\alpha\ln\beta$ for $\alpha$ and $\beta$ Algebraic Numbers.

See also Antilogarithm, Cologarithm, e, Exponential Function, Harmonic Logarithm, Lg, Ln, Log, Logarithmic Number, Napierian Logarithm, Natural Logarithm, Power


Abramowitz, M. and Stegun, C. 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.

Conway, J. H. and Guy, R. K. ``Logarithms.'' The Book of Numbers. New York: Springer-Verlag, pp. 248-252, 1996.

Beyer, W. H. ``Logarithms.'' CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 159-160, 1987.

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 $\ln(x)$.'' Ch. 25 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 225-232, 1987.

info prev up next book cdrom email home

© 1996-9 Eric W. Weisstein