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  2. Natural logarithm - Wikipedia

    en.wikipedia.org/wiki/Natural_logarithm

    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). The simplicity of this definition, which is matched in many ...

  3. List of logarithmic identities - Wikipedia

    en.wikipedia.org/wiki/List_of_logarithmic_identities

    ln(r) is the standard natural logarithm of the real number r. Arg(z) is the principal value of the arg function; its value is restricted to (−π, π]. It can be computed using Arg(x + iy) = atan2(y, x). Log(z) is the principal value of the complex logarithm function and has imaginary part in the range (−π, π].

  4. Logarithm - Wikipedia

    en.wikipedia.org/wiki/Logarithm

    Common logarithms (base 10), historically used in logarithm tables and slide rules, are a basic tool for measurement and computation in many areas of science and engineering; in these contexts log x still often means the base ten logarithm. [10] In mathematics log x usually refers to the natural logarithm (base e). [11]

  5. Logarithmic derivative - Wikipedia

    en.wikipedia.org/wiki/Logarithmic_derivative

    In summary, both derivatives and logarithms have a product rule, a reciprocal rule, a quotient rule, and a power rule (compare the list of logarithmic identities); each pair of rules is related through the logarithmic derivative.

  6. List of limits - Wikipedia

    en.wikipedia.org/wiki/List_of_limits

    4.1 Natural logarithms. 4.2 Logarithms to arbitrary bases. 5 Trigonometric functions. 6 Sums. ... All differentiation rules can also be reframed as rules involving ...

  7. Prime number theorem - Wikipedia

    en.wikipedia.org/wiki/Prime_number_theorem

    The first such distribution found is π(N) ~ ⁠ N / log(N) ⁠, where π(N) is the prime-counting function (the number of primes less than or equal to N) and log(N) is the natural logarithm of N. This means that for large enough N, the probability that a random integer not greater than N is prime is very close to 1 / log(N).

  8. Natural logarithm of 2 - Wikipedia

    en.wikipedia.org/wiki/Natural_logarithm_of_2

    In a third layer, the logarithms of rational numbers r = ⁠ a / b ⁠ are computed with ln(r) = ln(a) − ln(b), and logarithms of roots via ln n √ c = ⁠ 1 / n ⁠ ln(c).. The logarithm of 2 is useful in the sense that the powers of 2 are rather densely distributed; finding powers 2 i close to powers b j of other numbers b is comparatively easy, and series representations of ln(b) are ...

  9. Common logarithm - Wikipedia

    en.wikipedia.org/wiki/Common_logarithm

    The logarithm keys (log for base-10 and ln for base-e) on a typical scientific calculator. The advent of hand-held calculators largely eliminated the use of common logarithms as an aid to computation. The numerical value for logarithm to the base 10 can be calculated with the following identities: [5]