Definition:Logarithm
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Natural Logarithm
Positive Real Numbers
Let $x \in \R$ be a real number such that $x > 0$.
The (natural) logarithm of $x$ is defined as:
- $\displaystyle \ln x := \int_1^x \frac {\mathrm dt} t$
Complex Numbers
The complex natural logarithm of a complex value $z \in \C$ is written $\log \left({z}\right)$ (no base value) and is defined:
- $\log \left({z}\right) := \ln \left|{z}\right| + i \arg \left({z}\right)$
The principal branch of the complex logarithm is written and defined:
- $\operatorname{Log} \left({z}\right) := \ln \left|{z}\right| + i \operatorname{Arg} \left({z}\right)$
where $\arg \left({z}\right)$ is the continuous argument of $z$ and $\operatorname{Arg}\left({z}\right) = \arg \left({z}\right) \ \left({\bmod \left({2 \pi}\right)}\right)$.
General Logarithm
The natural logarithm function gives rise to the exponential function as follows:
- $x = \ln y \iff y = \exp x = e^x$
Thus the logarithm is the inverse of the exponential. It can also be independently shown that the logarithm function always exists without taking recourse to the fact that it is the inverse. For a proof see Existence of Logarithm.
Consider the general exponential function: $y = a^x = e^{x \ln a}$, where $a \in \R: a > 0$, as defined in Powers of Real Numbers.
As $\forall x \in \R: x \ln a \in \R$, and the nature of the exponential function (strictly increasing), we can define the function $\log_a y$:
- $x = \log_a y \iff y = a^x$
This is called the logarithm to the base $a$, or log base $a$.
When $a = e$, they are of course natural logarithms, and are sometimes called Napierian logarithms
When $a = 2$, the notation which is starting to be used for $\log_2 x$ is $\lg x$. This concept is becoming increasingly important in computer science.
The act of performing the $\log_a$ function is colloquially known as "taking logs".
Common Logarithms
When $a = 10$, the logarithms are common logarithms, sometimes called Briggsian Logarithms
References
- ↑ For John Napier.
- ↑ For Henry Briggs.
