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The discrete logarithm is the integer n solving the equation =, where x is an element of the group. Carrying out the exponentiation can be done efficiently, but the discrete logarithm is believed to be very hard to calculate in some groups.
The product logarithm Lambert W function plotted in the complex plane from −2 − 2i to 2 + 2i The graph of y = W(x) for real x < 6 and y > −4.The upper branch (blue) with y ≥ −1 is the graph of the function W 0 (principal branch), the lower branch (magenta) with y ≤ −1 is the graph of the function W −1.
Logarithms can be used to make calculations easier. 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.
The natural logarithm of a number is its logarithm to the base of the mathematical constant e, which is an irrational and transcendental number approximately equal to 2.718 281 828 459. [1] The natural logarithm of x is generally written as ln x, log e x, or sometimes, if the base e is implicit, simply log x.
This striking formula is one of the so-called explicit formulas of number theory, and is already suggestive of the result we wish to prove, since the term x (claimed to be the correct asymptotic order of ψ(x)) appears on the right-hand side, followed by (presumably) lower-order asymptotic terms.
As with other logarithms, the binary logarithm obeys the following equations, which can be used to simplify formulas that combine binary logarithms with multiplication or exponentiation: [9] log 2 x y = log 2 x + log 2 y {\displaystyle \log _{2}xy=\log _{2}x+\log _{2}y}
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]
Now, taking this derived formula, we can use Euler's formula to define the logarithm of a complex number. To do this, we also use the definition of the logarithm (as the inverse operator of exponentiation): a = e ln a , {\displaystyle a=e^{\ln a},} and that e a e b = e a + b , {\displaystyle e^{a}e^{b}=e^{a+b},} both valid for any complex ...