Logarithm
In 1614, John Napier published a book with a grand Latin title: Mirifici Logarithmorum Canonis Descriptio, the Description of the Wonderful Canon of Logarithms. The word he coined, logarithmus, was stitched together from the Greek logos and arithmos. Behind that ornate name sat a simple, radical idea. A logarithm is the exponent to which a fixed base must be raised to produce a given number. Why would navigators, surveyors, and astronomers seize on this so quickly that it spread across seventeenth-century Europe within years? What did it let them do that arithmetic alone could not? And how did a tool for turning multiplication into addition end up describing earthquakes, prime numbers, musical octaves, and the disorder of a box of gas? The answers run from a Belgian Jesuit chasing the area under a curve to Richard Feynman building a bit-by-bit algorithm at Los Alamos.
Addition, multiplication, and exponentiation are three of the most fundamental arithmetic operations, and each has an inverse. Subtraction undoes addition. Division undoes multiplication. The logarithm undoes exponentiation. When a base b is raised to a power y to give a value x, the logarithm runs the process backward, taking x and returning y. For a positive base b not equal to 1, the logarithm of a positive number x is the unique exponent that yields x. The whole apparatus rests on one identity that made logarithms historically irresistible. The logarithm of a product equals the sum of the logarithms of its factors, provided the base and both numbers are positive. That single rule turns a hard multiplication into an easy addition. Related identities followed: the logarithm of a ratio is a difference of logarithms, the logarithm of a p-th power is p times the logarithm, and the logarithm of a p-th root divides the logarithm by p. A change-of-base formula lets any logarithm be computed from logarithms in another base, which is why a scientific calculator only needs to know two.
Base 10 is called the common or decimal logarithm, and it slots neatly into the decimal number system. Its value is tied to the number of digits in a positive integer: the digit count is the smallest integer strictly bigger than the logarithm. For 5986, the base-10 logarithm is about 3.78, the next integer up is 4, and 5986 has four digits. The natural logarithm takes the constant e as its base, prized in mathematics and physics for its very simple derivative. The binary logarithm uses base 2 and runs through computer science, information theory, music theory, and photography. In information theory, the natural and binary logarithms correspond to nats and bits as fundamental units. In photography, rescaled base-2 logarithms measure exposure values, light levels, exposure times, lens apertures, and film speeds in stops. When the base is clear from context or simply does not matter, it is dropped, and the meaning shifts by field: base 10 in much of science and engineering, base e in mathematics, base 2 in computer science. The International Organization for Standardization even suggests notations for these bases.
The natural logarithm did not begin as a logarithm at all. It began as an attempt to find the area under a rectangular hyperbola, a quadrature. Archimedes had managed the parabola in his work The Quadrature of the Parabola in the third century BC, but the hyperbola resisted every effort. Grégoire de Saint-Vincent, a Belgian Jesuit living in Prague, finally published results on it in 1647. The breakthrough was a hidden pattern: the area behaved so that a geometric progression in the input produced an arithmetic progression in the output. That is the signature of a logarithm. A. A. de Sarasa recognized the link to the older tradition of logarithms, giving rise to the term hyperbolic logarithm as a synonym for the natural logarithm. The idea was soon taken up by Christiaan Huygens and James Gregory. Gottfried Wilhelm Leibniz adopted the notation in 1675 and the following year connected it to the integral. The natural logarithm can in fact be defined directly as the area between the x-axis and the curve of one over t, a definition that leans on nothing more than a simple reciprocal.
Henry Briggs compiled the first table of logarithms in 1617, just after Napier's invention, and made one decisive change: he used 10 as the base. His first table held the common logarithms of every integer from 1 to 1000, carried to a precision of 14 digits. The genius of the format lay in splitting each common logarithm into two parts, the integer characteristic and the fractional mantissa. Because numbers differing by factors of 10 share a mantissa, tables only needed to list mantissas, while a quick count of digits supplied the characteristic. A worker multiplying two numbers looked up both logarithms, added them, and looked up the antilogarithm of the sum. Pierre-Simon Laplace praised the result as an admirable artifice which, by reducing to a few days the labour of many months, doubles the life of the astronomer, and spares him the errors and disgust inseparable from long calculations. A second tool needed no table at all. The non-sliding Gunter's rule appeared shortly after Napier, and William Oughtred turned it into the slide rule, a pair of logarithmic scales that move against each other. Sliding one scale physically adds logarithms, so aligning a distance of 2 against a distance of 3 reads off their product, 6. The slide rule stayed essential for engineers and scientists until the 1970s.
The analytic heart of the logarithm is its derivative. The natural logarithm has the strikingly simple derivative of one over x, which makes it the unique antiderivative of that reciprocal taking the value 0 at 1. This simplicity is exactly what earned the natural logarithm its name and helps explain the importance of the constant e. The logarithm is also a transcendental function, a number or function that is not algebraic, in the company of pi and e. Computing logarithms by hand uses several routes. A Taylor series approximates the natural logarithm through partial sums that grow steadily more accurate. A faster series built on the inverse hyperbolic tangent converges more quickly, especially when the input is close to 1. For very high precision, the arithmetic-geometric mean takes over. Sasaki and Kanada showed in 1982 that this method was particularly fast for precisions between 400 and 1000 decimal places, using a formula due to Carl Friedrich Gauss. There is also a beautifully physical method. While at Los Alamos National Laboratory working on the Manhattan Project, Richard Feynman devised a bit-processing algorithm resembling long division, later used in the Connection Machine. It builds a product of distinct factors so that the logarithm reduces to simple addition from a table.
A logarithmic scale shrinks wide-ranging quantities into a manageable span, and the decibel is the textbook case. It is based on the common logarithm of ratios: ten times the logarithm of a power ratio, or twenty times the logarithm of a voltage ratio. Decibels quantify the attenuation or amplification of signals, the loudness of sounds, the absorbance of light, and the signal-to-noise ratio. Earthquakes are measured the same way, by the common logarithm of the energy released, in the moment magnitude and Richter scales. The numbers are vivid: a 5.0 quake releases 32 times the energy of a 4.0, and a 6.0 releases 1000 times as much. The brightness of stars, their apparent magnitude, is logarithmic too. In chemistry, the letter p marks the negative decimal logarithm, so pH measures the activity of hydronium ions. Neutral water has a hydronium activity of 10 to the minus 7 moles per litre, giving a pH of 7, while vinegar sits near 3. Semilog graphs scale one axis logarithmically, compressing the jump from 1 million to 1 trillion into the same span as 1 to 1 million. On a log-log graph, a power law appears as a straight line whose slope is the exponent.
Logarithms turn up wherever a problem splits in half. Binary search, hunting a value in a sorted list, needs on average a number of comparisons proportional to the logarithm of the list's length. Merge sort, which divides a list and merges the sorted halves, runs in time proportional to N times the logarithm of N. Human perception follows suit. Hick's law ties decision time to the logarithm of the number of choices, Fitts's law links movement time to a logarithm of distance over target size, and the Weber-Fechner law relates stimulus to sensation logarithmically. People with little mathematics education even tend to place numbers on a line by their logarithm, so 10 sits as close to 100 as 100 does to 1000. In number theory, the prime number theorem estimates how many primes lie below x using the natural logarithm. Benford's law predicts the leading digits of real-world data, so about 30 percent of values begin with 1 and 18 percent with 2, and auditors hunt deviations from it to flag fraudulent accounting. In music, base-2 logarithms convert frequency ratios into octaves, semitones, and cents, with the note A at 440 Hz, B-flat at 466 Hz, and B at 493 Hz. The concept reaches further still. The discrete logarithm, the multi-valued inverse of exponentiation in finite groups, is believed to be very hard to compute, and that very difficulty underpins the Diffie-Hellman key exchange that secures cryptographic keys over open channels.
Common questions
Who invented logarithms and when were they introduced?
John Napier introduced logarithms in 1614 in a book titled Mirifici Logarithmorum Canonis Descriptio, the Description of the Wonderful Canon of Logarithms. He coined the term logarithmus from the Greek words logos and arithmos.
What is a logarithm in mathematics?
A logarithm of a number is the exponent to which a fixed base must be raised to produce that number. As a single-variable function, the logarithm to base b is the inverse of exponentiation with base b.
What are the three most common logarithm bases?
The three common bases are 10, e, and 2. Base 10 is the common or decimal logarithm used in science and engineering, base e is the natural logarithm widespread in mathematics and physics, and base 2 is the binary logarithm used in computer science, information theory, music theory, and photography.
Why were logarithms important before calculators?
Logarithms turned multiplication and division into addition and subtraction, because the logarithm of a product is the sum of the logarithms of its factors. Using logarithm tables and slide rules, this replaced tedious multi-digit multiplication with table look-ups and simpler addition, greatly aiding astronomy, surveying, and navigation.
Who made the first table of logarithms?
Henry Briggs compiled the first table of logarithms in 1617, just after Napier's invention, using 10 as the base. His first table contained the common logarithms of all integers from 1 to 1000 with a precision of 14 digits.
How are logarithms used in measuring earthquakes and pH?
Earthquake strength is measured by the common logarithm of the energy released, used in the moment magnitude and Richter scales, so a 6.0 quake releases 1000 times the energy of a 4.0. In chemistry, pH is the negative decimal logarithm of hydronium ion activity, so neutral water has a pH of 7 and vinegar about 3.
What is the natural logarithm and why is it called natural?
The natural logarithm uses the constant e as its base and can be defined as the area under the curve of one over t. It is called natural because its derivative is the very simple expression one over x, which also explains the importance of the constant e.