There are, relatively speaking, no such simple solutions for factorials ; any combination of sums, products, powers, exponential functions, or logarithms with a fixed number of terms will not suffice to express n < nowiki >!</ nowiki >.

Logarithms can be defined to any positive base other than 1, not just e ; however logarithms in other bases differ only by a constant multiplier from the natural logarithm, and are usually defined in terms of the latter.

This work was extended by Alan Baker in the 1960s in his work on lower bounds for linear forms in any number of logarithms ( of algebraic numbers ).

The Wang LOCI-2 ( an earlier LOCI-1 was not a real product ) was introduced in 1965 and was probably the first desktop calculator capable of computing logarithms, quite an achievement for a machine without any integrated circuits.

The sum of two logarithms is equal to the logarithm of the product of their operands, i. e. for any base b > 0:

Paul Bourke credits Ben Rudiak-Gould with this description of how natural logarithms ( ln ()) can be used to represent any positive integer n as:

If the number of coin denominations is three or more, no explicit formula is known ; but, for any fixed number of coin denominations, there is an algorithm computing the Frobenius number in polynomial time ( in the logarithms of the coin denominations forming an input ).

Due to their utility in saving work in laborious multiplications and divisions with pen and paper, tables of base 10 logarithms were given in appendices of many books.

Thus, one has the appearance of logarithms in the simpler definition, previously given.

In mathematics, an elementary function is a function of one variable built from a finite number of exponentials, logarithms, constants, and nth roots through composition and combinations using the four elementary operations (+ – × ÷).

This table was actually based on Adriaan Vlacq's tables, but corrected a number of errors and extended the logarithms of trigonometric functions for the small angles.

It is useful because it gives us an idea of the melodic qualities of the tuning, and because if R is a rational number N / D, so is ( 3R + 1 )/( 5R + 2 ) or ( 3N + D )/( 5N + 2D ), which is the size of fifth in terms of logarithms base 2, and which immediately tells us what division of the octave we will have.

Rounding is almost unavoidable in many computations — especially when dividing two numbers in integer or fixed-point arithmetic ; when computing mathematical functions such as square roots, logarithms, and sines ; or when using a floating point representation with a fixed number of significant digits.

Such a table of " common logarithms " gave the logarithm, often to 4 or 5 decimal places, of each number in the left-hand column, which ran from 1 to 10 by small increments, perhaps 0. 01 or 0. 001.

Tables logarithms of trigonometric functions simplify hand calculations where a function of an angle must be multiplied by another number, as is often the case.

Laplace solved this problem for the case of rational functions, as he showed that the indefinite integral of a rational function is a rational function and a finite number of constant multiples of logarithms of rational functions.

In detail, multiplying estimates corresponds to adding their logarithms ; thus one obtains a sort of Wiener process or random walk on the logarithmic scale, which diffuses as ( in number of terms n ).

The fact that there are two logarithms ( log of a log ) in the limit for the Meissel – Mertens constant may be thought of as a consequence of the combination of the prime number theorem and the limit of the Euler – Mascheroni constant.

Their result involves the logarithm of a transcendental number instead of the logarithms of integers mentioned above.

He worked with the concept of ardhaccheda: the number of times a number could be divided by 2 ; effectively logarithms to base 2.

Zech logarithms are also called Jacobi Logarithm, after Jacobi who used them for number theoretic investigations

For sufficiently small finite fields, a table of Zech logarithms allows an especially efficient implementation of all finite field arithmetic in terms of a small number of integer addition / subtractions and table look-ups.

Earlier, in an era of hand calculation, suggested using sums of 50 digits taken from mathematical tables of logarithms as a form of random number generation ; if one assumes that each digit is random, then by the central limit theorem, these digit sums will have a random distribution closely approximating a Gaussian distribution.

A procedure called the Risch algorithm exists which is capable of determining if the integral of an elementary function ( function built from a finite number of exponentials, logarithms, constants, and nth roots through composition and combinations using the four elementary operations ) is elementary and returning it if it does.

* Simple built-in mathematical functions including trigonometric ( SIN, COS, TAN ), logarithms ( LOG, EXP ), square root ( SQR ), and random number generator ( RND )

** Euler's number, a transcendental number equal to 2. 71828182845 ... which is used as the base for natural logarithms

In number theory, the Pohlig – Hellman algorithm sometimes credited as the Silver – Pohlig – Hellman algorithm is a special-purpose algorithm for computing discrete logarithms in a multiplicative group whose order is a smooth integer.

Isaac Newton, at the age of twenty-three, industriously calculating logarithms `` to two and fifty places '' during the great plague year in England, 1665 ; ;

Some encryption schemes can be proven secure on the basis of the presumed difficulty of a mathematical problem, such as factoring the product of two large primes or computing discrete logarithms.

The logarithm must be taken to base e since the two terms following the logarithm are themselves base-e logarithms of expressions that are either factors of the density function or otherwise arise naturally.

A person can use logarithm tables to divide two numbers, by subtracting the two numbers ' logarithms, then looking up the antilogarithm of the result.

At the age of nine he learned to use a slide rule, followed shortly by the study of logarithms, and subsequently completed high school at Mulberry High School in two years.

After John Napier invented logarithms, and Edmund Gunter created the logarithmic scales ( lines, or rules ) upon which slide rules are based, it was Oughtred who first used two such scales sliding by one another to perform direct multiplication and division ; and he is credited as the inventor of the slide rule in 1622.

The equivalence of these two methods can be seen by recognizing that multiplication of sinusoids can be done by summing phases which are complex-number exponents of e, the base of natural logarithms.

The first two stages depend only on the generator g and prime modulus q, and find the discrete logarithms of a factor base of r small primes.

The fundamental difference between a calculator and computer is that a computer can be programmed in a way that allows the program to take different branches according to intermediate results, while calculators are pre-designed with specific functions such as addition, multiplication, and logarithms built in.

Practically simultaneously, Alan Baker proved an Baker's theorem on linear forms in logarithms of algebraic numbers which resolved the problem by a completely different method.

i. e., multiplication across all elements of a set is b to the power of the sum of all logarithms of the set's elements.

Borda was an enthusiast for the metric system and constructed tables of these logarithms starting in 1792 but their publication was delayed until after his death and only published in the Year 9 ( 1801 ) as Tables of Logarithms of sines, secants, and tangents, co-secants, co-sines, and co-tangents for the Quarter of the Circle divided into 100 degrees, the degree into 100 minutes, and the minute into 100 seconds to ten decimals, and including his tables of logarithms to 7 decimals from 10, 000 to 100, 000 with tables for obtaining results to 10 decimals.

There was only a need to include numbers between 1 and 10, since the logarithms of larger numbers were easily calculated.

These numbers are so large that they are typically only referred to using their logarithms.

The only important published extension of Vlacq's table was made by Mr. Sang in 1871, whose table contained the seven-place logarithms of all numbers below 200, 000.

This work, which contained the logarithms of all numbers up to 100, 000 to nineteen places, and of the numbers between 100, 000 and 200, 000 to twenty-four places, exists only in manuscript, " in seventeen enormous folios ," at the Observatory of Paris.

for some explicit function G, say built up from powers, logarithms and exponentials, that means only

This was a full-featured calculator that included not only standard " adding machine " functions but also powerful capabilities to handle floating-point numbers, trigonometric functions, logarithms, exponentiation, and square roots.