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The natural logarithm of a number is its logarithm to the base of the mathematical constant e, where e is an irrational and transcendental number equal to 2.718281828459. The natural logarithm of x is written as ln x, loge x, or sometimes, if the base e is implicit log x. Parentheses are sometimes added for clarity, giving loge, or log; this is done in particular when the argument to the logarithm is not a single symbol, to prevent ambiguity. The natural logarithm of x is the power to which e would have to be raised to equal x. For example, ln 7.5 is 2.0149... because e2.0149... = 7.5. The natural logarithm of e itself, ln e, is 1, because e1 = e, while the natural logarithm of 1 is 0, since e0 = 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. The simplicity of this definition, matched in many other formulas involving the natural logarithm, leads to the term "natural"; the definition of the natural logarithm can be extended to give logarithm values for negative numbers and for all non-zero complex numbers, although this leads to a multi-valued function: see Complex logarithm.

The natural logarithm function, if considered as a real-valued function of a real variable, is the inverse function of the exponential function, leading to the identities: e ln ⁡ x = x if x > 0, ln ⁡ e x = x. Like all logarithms, the natural logarithm maps multiplication into addition: ln ⁡ x y = ln ⁡ x + ln ⁡ y. Logarithms can be defined for any positive base other than 1, not only e. However, logarithms in other bases differ only by a constant multiplier from the natural logarithm, can be defined in terms of the latter. For instance, the base-2 logarithm is equal to the natural logarithm divided by ln 2, the natural logarithm of 2. Logarithms are useful for solving equations in which the unknown appears as the exponent of some other quantity. For example, logarithms are used to solve for the half-life, decay constant, or unknown time in exponential decay problems, they are important in many branches of mathematics and the sciences and are used in finance to solve problems involving compound interest.

The concept of the natural logarithm was worked out by Gregoire de Saint-Vincent and Alphonse Antonio de Sarasa before 1649. Their work involved quadrature of the hyperbola with equation xy = 1 by determination of the area of hyperbolic sectors, their solution generated the requisite "hyperbolic logarithm" function having properties now associated with the natural logarithm. An early mention of the natural logarithm was by Nicholas Mercator in his work Logarithmotechnia published in 1668, although the mathematics teacher John Speidell had in 1619 compiled a table of what in fact were natural logarithms, it has been said that Speidell's logarithms were to the base e, but this is not true due to complications with the values being expressed as integers. The notations ln x and loge x both refer unambiguously to the natural logarithm of x, log x without an explicit base may refer to the natural logarithm; this usage is common in mathematics and some scientific contexts as well as in many programming languages.

In some other contexts, log x can be used to denote the common logarithm. The natural logarithm can be defined in several equivalent ways; the natural logarithm of a positive, real number a may be defined as the area under the graph of the hyperbola with equation y = 1/x between x = 1 and x = a. This is the integral ln ⁡ a = ∫ 1 a 1 x d x. If a is less than 1, this area is considered to be negative; this function is a logarithm because it satisfies the fundamental property of a logarithm: ln ⁡ = ln ⁡ a + ln ⁡ b. This can be demonstrated by splitting the integral that defines ln ab into two parts and making the variable substitution x = at in the second part, as follows: ln ⁡ a b = ∫ 1 a b 1 x d x = ∫ 1 a 1 x d x + ∫ a a b 1 x d x = ∫ 1 a 1 x d x + ∫ 1 b 1 a t a d t

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Sheriff is the self-titled debut album and only album by Canadian band Sheriff released in 1982 on Capitol Records. The single "When I'm with You" jumped to the top of the Billboard Hot 100 in 1989, seven years after its release. All songs were written except where noted. Freddy Curci – lead vocals Steve DeMarchiguitar, backing vocals Wolf D. Hassel – bass, backing vocals Arnold Lannikeyboards, piano, backing vocals Rob Elliottdrums Executive Producers: John Victor and Helen Victor Engineered by Greg Roberts, assisted by Alyx Skriabow Pre-Production at Perceptions Recording Studio, Ontario Mixed at Electric Lady Studios, New York City Mix Engineered by Dave Wittman, assisted by Ed Garcia and Michel Sauvage Mastered at Masterdisk, New York City, by Bob Ludwig Album - Billboard Single - Billboard