In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the 3rd power: 1000 = 103 = 10 × 10 × 10. More generally, if x = by, then y is the logarithm of x to base b, written logb x, so log10 1000 = 3. As a single-variable function, the logarithm to base b is the inverse of exponentiation with base b.

Logarithms were introduced by John Napier in 1614 as a means of simplifying calculations. They were rapidly adopted by navigators, scientists, engineers, surveyors, and others to perform high-accuracy computations more easily. Using logarithm tables, tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition. This is possible because the logarithm of a product is the sum of the logarithms of the factors:

log_b(xy) = log_b ⁡x + log_b ⁡y ,

provided that b, x and y are all positive and b ≠ 1. The slide rule, also based on logarithms, allows quick calculations without tables, but at lower precision.

The common logarithm of a number is the index of that power of ten which equals the number. Speaking of a number as requiring so many figures is a rough allusion to common logarithm, and was referred to by Archimedes as the “order of a number”. The first real logarithms were heuristic methods to turn multiplication into addition, thus facilitating rapid computation.

https://en.m.wikipedia.org/wiki/Logarithm

John Napier of Merchiston Latinized as Ioannes Neper; 1 February 1550 – 4 April 1617), nicknamed Marvellous Merchiston, was a Scottish landowner known as a mathematician, physicist, and astronomer. He was the 8th Laird of Merchiston.

John Napier is best known as the discoverer of logarithms. He also invented the so-called

"Napier's bones"

Napier’s bones is a manually operated calculating device created by John Napier of Merchiston, Scotland for the calculation of products and quotients of numbers. The method was based on lattice multiplication, and also called rabdology, a word invented by Napier. Napier published his version in 1617. It was printed in Edinburgh and dedicated to his patron Alexander Seton.

https://en.m.wikipedia.org/wiki/Napier's_bones

::: and popularised the use of the decimal point in arithmetic and mathematics.

Napier’s birthplace, Merchiston Tower in Edinburgh, is now part of the facilities of Edinburgh Napier University. There is a memorial to him at St Cuthbert’s Parish Church at the west end of Princes Street Gardens in Edinburgh.

https://en.m.wikipedia.org/wiki/John_Napier

John Napier was a Scottish scholar who is best known for his invention of logarithms, but other mathematical contributions include a mnemonic for formulas used in solving spherical triangles and two formulas known as Napier’s analogies.

https://mathshistory.st-andrews.ac.uk/Biographies/Napier/

How to Write it

We write it like this:

log₂(8) = 3

So these two things are the same:

The number we multiply is called the “base”, so we can say:

• “the logarithm of 8 with base 2 is 3”
• or “log base 2 of 8 is 3”
• or “the base-2 log of 8 is 3”

Notice we are dealing with three numbers:

• the base: the number we are multiplying (a “2” in the example above)
• how often to use it in a multiplication (3 times, which is the logarithm)
• The number we want to get (an “8”)

Example: What is log₅(625) … ?

We are asking “how many 5s need to be multiplied together to get 625?”

5 × 5 × 5 × 5 = 625, so we need 4 of the 5s

Answer: log₅(625) = 4

https://www.mathsisfun.com/algebra/logarithms.html

Before Logarithms:

The late sixteenth century saw unprecedented development in many scientific fields; notably, observational astronomy, long-distance navigation, and geodesy science, or efforts to measure and represent the earth. These endeavors required much from mathematics. For the most part, their foundation was trigonometry, and trigonometric tables, identities, and related calculation were the subject of intensive enterprise. Typically, trigonometric functions were based on non-unity radii, such as R=10,000,000,

to ensure precise integer output.* Reducing the calculation burden that resulted from dealing with such large numbers for practitioners in these applied disciplines, and with it, the errors that inevitably crept into the results, became a prime objective for mathematicians. As a result, much energy and scholarly effort were directed towards the art of computation.

Accordingly, techniques that could bypass lengthy processes, such as long multiplications or divisions, were explored. Of particular interest were those that replaced these processes with equivalent additions and subtractions. One method originating in the late sixteenth century that was used extensively to save computation was the technique called prosthaphaeresis, a compound constructed from the Greek terms prosthesis (addition) and aphaeresis (subtraction). This relation transformed long multiplications and divisions into additions and subtractions via trigonometric identities, such as:

2cos(A)cos(B) = cos(A+B) + cos(A−B).

When one needed the product of two numbers x and y, for example, trigonometric tables would be consulted to find A and B such that:

x=cos(A) and y=cos(B).

With A and B determined, cos(A+B) and cos(A−B)

could be read from the table and half of the sum taken to find the original product in question. Thus the long multiplication of two numbers could be replaced by table look-up, addition, and halving. Such rules were recognized as early as the beginning of the sixteenth century by Johannes Werner in 1510, but their application specifically for multiplication first appeared in print in 1588 in a work by Nicolai Reymers Ursus (Thoren, 1988). Christopher Clavius extended the methods of prosthaphaeresis, of which examples can be found in his 1593 Astrolabium (Smith, 1959, p. 455).

Finally, with the scientific community focused on developing more powerful computational methods, the desire to capture symbolically essential mathematical ideas behind these developments was also growing. In the fifteenth and sixteenth centuries, mathematicians such as Nicolas Chuquet (c. 1430–1487) and Michael Stifel (c. 1487–1567) turned their attention to the relationship between arithmetic and geometric sequences while working to construct notation to express an exponential relationship. The focus on mathematical symbolism in centuries prior and the growing attention to notation–particularly the experimentation with different versions of exponent notation–played a critical role in the recognition and clarification of such a relationship. Now the mathematical connection between a geometric and an arithmetic sequence could be made all the more apparent by symbolically capturing these sequences as successive exponential powers of a given number and the exponents themselves, respectively (see Figure 6). The work on the relationships between sequences was mathematically important per se, but was equally significant for providing the inspiration for the development of the logarithmic relation.

* Note: Modern trigonometry is essentially based on triangles inscribed in a unit circle; that is, a circle with radius R=1. Early practitioners used circles with various values for the radius. The relationship between the modern sine function and a sine or half-chord in a circle of radius R is given by Sinθ = R sinθ, where the modern sine function has a lower case ‘s’ and the pre-modern sine an upper case ‘S’.

https://old.maa.org/press/periodicals/convergence/logarithms-the-early-history-of-a-familiar-function-before-logarithms-the-computational-demands-of

John Napier Introduces Logarithms

In such conditions, it is hardly surprising that many mathematicians were acutely aware of the issues of computation and were dedicated to relieving practitioners of the calculation burden. In particular, the Scottish mathematician John Napier was famous for his devices to assist with computation. He invented a well-known mathematical artifact, the ingenious numbering rods more quaintly known as “Napier’s bones,” that offered mechanical means for facilitating computation. (For additional information on “Napier’s bones,” see the article, “John Napier: His Life, His Logs, and His Bones” (2006).) In addition, Napier recognized the potential of the recent developments in mathematics, particularly those of prosthaphaeresis, decimal fractions, and symbolic index arithmetic, to tackle the issue of reducing computation. He appreciated that, for the most part, practitioners who had laborious computations generally did them in the context of trigonometry. Therefore, as well as developing the logarithmic relation, Napier set it in a trigonometric context so it would be even more relevant.

Napier first published his work on logarithms in 1614 under the title Mirifici logarithmorum canonis descriptio, which translates literally as A Description of the Wonderful Table of Logarithms. Indeed, the very title Napier selected reveals his high ambitions for this technique—the provision of tables based on a relation that would be nothing short of “wonder-working” for practitioners. As well as providing a short overview of the mathematical details, Napier gave technical expression to his concept. He coined a term from the two ancient Greek terms logos, meaning proportion, and arithmos, meaning number; compounding them to produce the word “logarithm.” Napier used this word as well as the designations “natural” and “artificial” for numbers and their logarithms, respectively, in his text.

Despite the obvious connection with the existing techniques of prosthaphaeresis and sequences, Napier grounded his conception of the logarithm in a kinematic framework. The motivation behind this approach is still not well understood by historians of mathematics. Napier imagined two particles traveling along two parallel lines. The first line was of infinite length and the second of a fixed length (see Figures 2 and 3). Napier imagined the two particles to start from the same (horizontal) position at the same time with the same velocity. The first particle he set in uniform motion on the line of infinite length so that it covered equal distances in equal times. The second particle he set in motion on the finite line segment so that its velocity was proportional to the distance remaining from the particle to the fixed terminal point of the line segment.

Figure 2. Napier’s two parallel lines with moving particles (Image used courtesy of Landmarks of Science Series, NewsBank-Readex)

More specifically, at any moment the distance not yet covered on the second (finite) line was the sine and the traversed distance on the first (infinite) line was the logarithm of the sine. This had the result that as the sines decreased, Napier’s logarithms increased. Furthermore, the sines decreased in geometric proportion, and the logarithms increased in arithmetic proportion. We can summarize Napier’s explanation as follows (Descriptio I, 1 (p. 4); see Figure 3):

AC = log_nap(γω) where γω = Sinθ₁

AD = log_nap(δω) where δω = Sinθ₂

AE = log_nap(ϵω) where ϵω= Sinθ₃

and so on, so that, more generally: x = Sin(θ)

y = log_nap(x)

where log_nap has been used to distinguish Napier’s particular understanding of the logarithm concept from the modern one.

Figure 3. The relation between the two lines and the logs and sines

Napier generated numerical entries for a table embodying this relationship. He arranged his table by taking increments of arc θ minute by minute, then listing the sine of each minute of arc, and then its corresponding logarithm. However in terms of the way he actually computed these entries, he would have in fact worked in the opposite manner, generating the logarithms first and then choosing those that corresponded to a sine of an arc, which accordingly formed the argument. For example, he would have computed values that appear in the first column of Table 1 via the relation:

Table 1. Napier’s logarithms

The values in the first column (in bold) that corresponded to the Sines of the minutes of arcs (third column) were extracted, along with their accompanying logarithms (column 2) and arranged in the table. The appropriate values from Table 1 can be seen in rows one to six of the last three columns in Figure 4. Napier tabulated his logarithms from 0∘ to 45∘ in minutes of arc, and by symmetry provided values for the entire first quadrant. The excerpt in Figure 4 gives the first half of the first degree and, by symmetry, on the right the last half of the eighty-ninth degree.

To complete the tables, Napier computed almost ten million entries from which he selected the appropriate values. Napier himself reckoned that computing this many entries had taken him twenty years, which would put the beginning of his endeavors as far back as 1594.

Figure 4. The first page of Napier’s tables

(Image used courtesy of Landmarks of Science Series, NewsBank-Readex)

Napier frequently demonstrated the benefits of his method. For example, he worked through a problem involving the computation of mean proportionals, sometimes known as the geometric mean. He reviewed the usual way in which this would have been computed, and pointed out that his technique using logarithms not only finds the answer “earlier” (that is, faster!), but also uses only one addition and one division by two! He stated:

“Let the extremes 1000000 and 500000 bee given, and let the meane proportionall be sought: that commonly is found by multiplying the extreames given, one by another, and extracting the square root of the product. But we finde it earlier thus; We adde the Logarithme of the extreames 0 and 693147, the summe whereof is 693147 which we divide by 2 and the quotient 346573 shall be the Logar. of the middle proportionall desired. By which the middle proportionall 707107, and his arch 45 degrees are found as before… found by addition onely, and division by two. (Book I, 5 (p. 25), as translated by Edward Wright)”

In order to find the mean proportional by traditional methods, Napier observed that one has to compute the product and then take the square root; that is:

√(1000000×500000) = √(500000000000) ≈ 707106.78

This method involves the multiplication of two large numbers and a lengthy square-root extraction. As an alternative, Napier proposed (with computations to 6 significant figures):

log_nap(1000000)+log_nap(500000)=0+693147=693147

693147÷2 = 346573 to 6 significant figures

⇒mean proportional = 707107, as required,

which he rightly deemed was much simpler to compute.

https://old.maa.org/press/periodicals/convergence/logarithms-the-early-history-of-a-familiar-function-john-napier-introduces-logarithms

Henry Briggs and the Common Logarithm

Shortly after Napier’s publication, English mathematician Henry Briggs (1561–1630) refined and popularized the concept of logarithms. Briggs collaborated with Napier and proposed the use of base-10 logarithms, also known as common logarithms. In 1617, Briggs published Logarithmorum Chilias Prima, containing the first table of base-10 logarithms.

Briggs’ base-10 system was more intuitive and practical for everyday calculations, as it aligned with the decimal system widely used in Europe. This refinement made logarithms accessible to a broader audience, including scientists, engineers, and navigators.

The Logarithmic Scale: Slide Rules and Early Calculators

One of the earliest applications of logarithms was the development of the slide rule. In 1622, English mathematician William Oughtred invented the circular slide rule, which utilized logarithmic scales for rapid calculations. By the mid-17th century, linear slide rules became common tools for scientists, engineers, and students.

The slide rule remained an essential computational device for over 300 years, until the advent of electronic calculators in the mid-20th century. Its reliance on logarithmic principles demonstrates the enduring utility of logarithms in simplifying calculations.

Logarithms in Astronomy and Navigation

Logarithms played a crucial role in advancing astronomy and navigation during the 17th and 18th centuries. Astronomers like Johannes Kepler and Isaac Newton relied on logarithmic tables to perform complex calculations related to planetary motion and celestial mechanics. By reducing the computational burden, logarithms enabled astronomers to make precise predictions and refine their models of the universe.

Navigators also benefited from logarithms, particularly in determining longitude and calculating distances at sea. The efficiency of logarithmic tables allowed mariners to improve their accuracy in charting courses and conducting explorations.

https://www.historymath.com/logarithms/

The Slide rule

This is a picture of a basic beginner’s slide rule for various math operations including mutiplication/division and square/squareroot:

Components of A Slide Rule

The slide rule is actually made of three bars that are fixed together. The sliding center bar is sandwiched by the outer bars which are fixed with respect to each other. The metal “window” is inserted over the slide rule to act as a place holder. A cursor is fixed in the center of the “window” to allow for accurate readings.

The scales (A-D) are labeled on the left-hand side of the slide rule. The number of scales on a slide rule vary depending on the number of mathematical functions the slide rule can perform. Multiplication and division are performed using the C and D scales. Square and square root are performed with the A and B scales. The numbers are marked according to a logarithmic scale. Therefore, the first number on the slide rule scale (also called the index) is 1 because the log of zero is one.

To know how it works please read the full page

Notice that on this scale the distance between the divisions is decreasing. This is a characteristic of a log scale. A logarithm relates one number to another number much like a mathematical function. The log of a number, to the base 10, is defined by:

The “magic” of the slide rule is actually based on a mathematical logarithmic relation:

These relations made it possible to perform multiplication and division using addition and subtraction. Before the slide rule, the product of two numbers were found by looking up their respective logs and adding them together, then finding the number whose log is the sum, also called the inverse log.

The slide rule made its first appearance in the late 17th century. The slide rule made it easier to utilize the log relations by developing a number line on which the displacement of the numbers were proportional to their logs. The slide rule eased the addition of the two logarithmic displacements of the numbers, thus assisting with multiplication and division in calculations.

LIMITING PHYSICS:

The accuracy of the calculations made with a slide rule depends on the accuracy with which the user can read the numbers off the scale. More divisions allow for more decimal places which means increased accuracy.

https://web.mit.edu/2.972/www/reports/slide_rule/slide_rule.html