Logarithm definitions and example sentences on Wordnik

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noun Mathematics The power to which a base, such as 10, must be raised to produce a given number. If n^x = a, the logarithm of a, with n as the base, is x; symbolically, log_n a = x. For example, 10³ = 1,000; therefore, log₁₀ 1,000 = 3. The kinds most often used are the common logarithm (base 10), the natural logarithm (base e), and the binary logarithm (base 2).

An artificial number, or number used in computation, belonging to a series (or system of logarithms) having the following properties: First, every natural or positive number, integral or fractional, has a logarithm in each system of logarithms; and conversely, every logarithm belongs to a natural number, called its antilogarithm. Second, in each system of logarithms, the logarithms corresponding to any geometrical progression of natural numbers are in arithmetical progression: that is, if each natural number of the series is obtained from the preceding one by multiplying a constant factor into this preceding one, then each logarithm may be obtained from the preceding one by adding a constant increment or subtracting a constant decrement. This is shown, for the system of Napier's logarithms, in the following table. It must be said that logarithms are, in general, irrational numbers, and their values can only be expressed approximately, being carried to some finite number of decimal places. Owing to the neglected places, it will often happen that the difference between two logarithms, obtained by subtracting the approximate value of one from that of the other, is in error by 1 in the last decimal place.

Natural numbers.	Logarithms (Napier's system).	Successive differences.
45.3999	123025851	23025851
453.9993	100000000	23025851
4539.9930	76974149	23025851
45399.9298	53948298	23025851
453999.2976	30922447	23025851
4539992.9763	7896596	23025851
45399929.7625	-15129255

It will thus be seen that if four numbers, A, B, C, D, are in proportion, so that A:B = C:D, then their four logarithms satisfy the equation, log A—log B = log C—log D; so that, to work the rule of three with logarithms, we simply substitute for each number its logarithm and proceed as usual, only that in every case we perform addition instead of multiplication and subtraction instead of division; and the result is the logarithm of the answer.

As now understood, a system of logarithms, besides the two essential characters set forth above, has a third, namely that the logarithm of 1 is 0. This being admitted, a simpler definition can be given of the logarithm, viz.: a logarithm is the exponent of the power to which a number constant for each system, and called the base of the system, must be raised in order to produce the natural number, or antilogarithm. Thus (base)^{log x} = x. At the time logarithms were invented fractional exponents had not been thought of, and even decimals, as we conceive them, were little used, the decimal point not having yet appeared; consequently, the last definition of the logarithm, which is now the usual one, was not at first possible. With logarithms in the modern sense, the rule for solving proportions still holds, but is secondary to the following fundamental rule: The sum of the logarithms of several numbers is the logarithm of the continued product of those numbers. For example, let it be required to determine the circumference of the earth in inches, knowing that its radius is 3958 miles. We take out from a table of logarithms the logarithms of all the numbers which have to be multiplied together, as follows:
The sum of these logarithms is 9.1974808, which we find by the table to be the logarithm of a number comprised between 1575690000 and 1575091000. To obtain a closer approximation, we should have to carry the logarithms to more places of decimals; but this would be useless, since the radius of the earth is only given to the nearest mile. From this fundamental rule several subsidiary rules follow as corollaries. Thus, to divide one number by another, subtract the logarithm of the divisor from that of the dividend, and the antilogarithm of the remainder is the quotient; to take the reciprocal of a number, change the sign of the logarithm, and the antilogarithm of the result is the reciprocal; to raise a number to any power, multiply the logarithm of the base by the exponent of the power, and the antilogarithm of the product is the power sought; to extract any root of a number, divide the logarithm of that number by the index of the root, and the antilogarithm of the quotient is the root sought. For example, what is the amount of $1 at interest at 6 per cent. compounding yearly for 1,000 years? We must here raise 1.06 to the thousandth power. The common logarithm of 1.06 is 0.0253058653; 1,000 times this is 25.3058653, which is the logarithm of 2022384 followed by 19 ciphers, or say 20 quadrillions 223840 trillions, in the English numeration. To give an idea of the advantage of logarithms in trigonometrical calculations, it may be mentioned that to find the altitude of the sun from its hour-angle and declination with logarithms requires seven numbers to be taken out of the tables and two additions to be performed, while the solution of the same problem with a table of natural sines requires, as before, the taking out of seven numbers from the tables, and besides eight additions and two halvings. There are two systems of logarithms in common use, the hyperbolic, natural, or Napicrian or Neperian (not Napier's own) logarithms in analysis, and common, decimal, or Briggsian logarithms in ordinary computations. The base of the system of hyperbolic logarithms is 2.718281828459. This kind of logarithm derives its name from its measuring the area between the equilateral hyperbola, an ordinate, and the axes of coordinates when these are the asymptotes; but the chief characteristic of the system is that, x being any number less than unity, Thus, the hyperbolic logarithm of 1.1 is calculated as follows:

15 more from this dictionary »

In the custom of the country the nose of an enemy stands as the logarithm of his head, which is inconvenient of transportation in number; and, though the Prince had forbidden the mutilation of the dead, it was impossible to enforce the prohibition out of Montenegro, and this was the only proof of the actual fruits of victory permitted by the circumstances. — The Autobiography of a Journalist, Volume II
* In [[logarithm]] s, the base is the quantity raised to the power of the logarithm to return the given number. — Citizendium, the Citizens' Compendium - Recent changes [en]
Discrete logarithm - are complete enough for approval. — Citizendium, the Citizens' Compendium - Recent changes [en]
E is the product of five factors, the four leading ones being: pi, rho, L and the square of A; and the product of these four is multiplied by the mathematical function called the natural logarithm (LOGe) of the ratio Ro / Ri. — CounterPunch
Numbers like e (the base of the natural logarithm, which may be defined by means of an infinite sum or infinite product), pi (the circumference of a circle divided by the diameter - which may also be defined in terms of an infinite sum), or phi (the ratio of the golden mean - one half of the sum of one and the square root of five). — RealClimate

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