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Exponentiation occurs in many areas of mathematics and its inverse function is often referred to as the logarithm. For example, the logarithm of a matrix is the (multi-valued) inverse function of the matrix exponential. [97] Another example is the p-adic logarithm, the inverse function of the p-adic exponential.
The LSE function is often encountered when the usual arithmetic computations are performed on a logarithmic scale, as in log probability. [5]Similar to multiplication operations in linear-scale becoming simple additions in log-scale, an addition operation in linear-scale becomes the LSE in log-scale:
For example, two numbers can be multiplied just by using a logarithm table and adding. These are often known as logarithmic properties, which are documented in the table below. [2] The first three operations below assume that x = b c and/or y = b d, so that log b (x) = c and log b (y) = d. Derivations also use the log definitions x = b log b (x ...
The definition of e x as the exponential function allows defining b x for every positive real numbers b, in terms of exponential and logarithm function. Specifically, the fact that the natural logarithm ln(x) is the inverse of the exponential function e x means that one has = () =
The exponential of a matrix A is defined by =!. Given a matrix B, another matrix A is said to be a matrix logarithm of B if e A = B.. Because the exponential function is not bijective for complex numbers (e.g. = =), numbers can have multiple complex logarithms, and as a consequence of this, some matrices may have more than one logarithm, as explained below.
Logarithmic growth is the inverse of exponential growth and is very slow. [2] A familiar example of logarithmic growth is a number, N, in positional notation, which grows as log b (N), where b is the base of the number system used, e.g. 10 for decimal arithmetic. [3] In more advanced mathematics, the partial sums of the harmonic series
For example, ln 7.5 is 2.0149..., because e 2.0149... = 7.5. The natural logarithm of e itself, ln e, is 1, because e 1 = e, while the natural logarithm of 1 is 0, since e 0 = 1. The natural logarithm can be defined for any positive real number a as the area under the curve y = 1/x from 1 to a [4] (with the area being negative when 0 < a < 1 ...
In mathematics, the common logarithm (aka "standard logarithm") is the logarithm with base 10. [1] It is also known as the decadic logarithm , the decimal logarithm and the Briggsian logarithm . The name "Briggsian logarithm" is in honor of the British mathematician Henry Briggs who conceived of and developed the values for the "common logarithm".