1.
Asterisk
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An asterisk is a typographical symbol or glyph. It is so called because it resembles a conventional image of a star, computer scientists and mathematicians often vocalize it as star. In English, an asterisk is usually five-pointed in sans-serif typefaces, six-pointed in serif typefaces and it can be used as censorship. It is also used on the Internet to correct ones spelling, the asterisk is derived from the need of the printers of family trees in feudal times for a symbol to indicate date of birth. The original shape was seven-armed, each arm like a shooting from the center. In computer science, the asterisk is used as a wildcard character, or to denote pointers, repetition. Origin Adamantius is known to have used the asteriskos to mark missing Hebrew lines from his Hexapla. The asterisk evolved in shape over time, but its meaning as a used to correct defects remained. In the Middle Ages, the asterisk was used to emphasize a part of text. However, an asterisk was not always used, one hypothesis to the origin of the asterisk is that it stems from the five thousand year old Sumerian character dingir,
2.
Circle
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A circle is a simple closed shape in Euclidean geometry. The distance between any of the points and the centre is called the radius, a circle is a simple closed curve which divides the plane into two regions, an interior and an exterior. Annulus, the object, the region bounded by two concentric circles. Arc, any connected part of the circle, centre, the point equidistant from the points on the circle. Chord, a segment whose endpoints lie on the circle. Circumference, the length of one circuit along the circle, or the distance around the circle and it is a special case of a chord, namely the longest chord, and it is twice the radius. Disc, the region of the bounded by a circle. Lens, the intersection of two discs, passant, a coplanar straight line that does not touch the circle. Radius, a line segment joining the centre of the circle to any point on the circle itself, or the length of such a segment, sector, a region bounded by two radii and an arc lying between the radii. Segment, a region, not containing the centre, bounded by a chord, secant, an extended chord, a coplanar straight line cutting the circle at two points. Semicircle, an arc that extends from one of a diameters endpoints to the other, in non-technical common usage it may mean the diameter, arc, and its interior, a two dimensional region, that is technically called a half-disc. A half-disc is a case of a segment, namely the largest one. Tangent, a straight line that touches the circle at a single point. The word circle derives from the Greek κίρκος/κύκλος, itself a metathesis of the Homeric Greek κρίκος, the origins of the words circus and circuit are closely related. The circle has been known since before the beginning of recorded history, natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with related inventions such as gears, in mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. Some highlights in the history of the circle are,1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as a value of π.300 BCE – Book 3 of Euclids Elements deals with the properties of circles
3.
Electrical engineering
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Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics, and electromagnetism. This field first became an occupation in the later half of the 19th century after commercialization of the electric telegraph, the telephone. Subsequently, broadcasting and recording media made electronics part of daily life, the invention of the transistor, and later the integrated circuit, brought down the cost of electronics to the point they can be used in almost any household object. Electrical engineers typically hold a degree in engineering or electronic engineering. Practicing engineers may have professional certification and be members of a professional body, such bodies include the Institute of Electrical and Electronics Engineers and the Institution of Engineering and Technology. Electrical engineers work in a wide range of industries and the skills required are likewise variable. These range from basic circuit theory to the management skills required of a project manager, the tools and equipment that an individual engineer may need are similarly variable, ranging from a simple voltmeter to a top end analyzer to sophisticated design and manufacturing software. Electricity has been a subject of scientific interest since at least the early 17th century and he also designed the versorium, a device that detected the presence of statically charged objects. In the 19th century, research into the subject started to intensify, Electrical engineering became a profession in the later 19th century. Practitioners had created an electric telegraph network and the first professional electrical engineering institutions were founded in the UK. Over 50 years later, he joined the new Society of Telegraph Engineers where he was regarded by other members as the first of their cohort, Practical applications and advances in such fields created an increasing need for standardised units of measure. They led to the standardization of the units volt, ampere, coulomb, ohm, farad. This was achieved at a conference in Chicago in 1893. During these years, the study of electricity was considered to be a subfield of physics. Thats because early electrical technology was electromechanical in nature, the Technische Universität Darmstadt founded the worlds first department of electrical engineering in 1882. The first course in engineering was taught in 1883 in Cornell’s Sibley College of Mechanical Engineering. It was not until about 1885 that Cornell President Andrew Dickson White established the first Department of Electrical Engineering in the United States, in the same year, University College London founded the first chair of electrical engineering in Great Britain. Professor Mendell P. Weinbach at University of Missouri soon followed suit by establishing the engineering department in 1886
4.
Film speed
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Film speed is the measure of a photographic films sensitivity to light, determined by sensitometry and measured on various numerical scales, the most recent being the ISO system. A closely related ISO system is used to measure the sensitivity of digital imaging systems, highly sensitive films are correspondingly termed fast films. In both digital and film photography, the reduction of exposure corresponding to use of higher sensitivities generally leads to reduced image quality, in short, the higher the sensitivity, the grainier the image will be. Ultimately sensitivity is limited by the efficiency of the film or sensor. The speed of the emulsion was then expressed in degrees Warnerke corresponding with the last number visible on the plate after development. Each number represented an increase of 1/3 in speed, typical speeds were between 10° and 25° Warnerke at the time. The concept, however, was built upon in 1900 by Henry Chapman Jones in the development of his plate tester. In their system, speed numbers were inversely proportional to the exposure required, for example, an emulsion rated at 250 H&D would require ten times the exposure of an emulsion rated at 2500 H&D. The methods to determine the sensitivity were later modified in 1925, the H&D system was officially accepted as a standard in the former Soviet Union from 1928 until September 1951, when it was superseded by GOST 2817-50. The Scheinergrade system was devised by the German astronomer Julius Scheiner in 1894 originally as a method of comparing the speeds of plates used for astronomical photography, Scheiners system rated the speed of a plate by the least exposure to produce a visible darkening upon development. ≈2 The system was extended to cover larger ranges and some of its practical shortcomings were addressed by the Austrian scientist Josef Maria Eder. Scheiners system was abandoned in Germany, when the standardized DIN system was introduced in 1934. In various forms, it continued to be in use in other countries for some time. The DIN system, officially DIN standard 4512 by Deutsches Institut für Normung, was published in January 1934, International Congress of Photography held in Dresden from August 3 to 8,1931. The DIN system was inspired by Scheiners system, but the sensitivities were represented as the base 10 logarithm of the sensitivity multiplied by 10, similar to decibels. Thus an increase of 20° represented an increase in sensitivity. ≈3 /10 As in the Scheiner system, speeds were expressed in degrees, originally the sensitivity was written as a fraction with tenths, where the resultant value 1.8 represented the relative base 10 logarithm of the speed. Tenths were later abandoned with DIN4512, 1957-11, and the example above would be written as 18° DIN, the degree symbol was finally dropped with DIN4512, 1961-10
5.
Logarithm
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In mathematics, the logarithm is the inverse operation to exponentiation. That means the logarithm of a number is the exponent to which another fixed number, in simple cases the logarithm counts factors in multiplication. For example, the base 10 logarithm of 1000 is 3, the logarithm of x to base b, denoted logb, is the unique real number y such that by = x. For example, log2 =6, as 64 =26, the logarithm to base 10 is called the common logarithm and has many applications in science and engineering. The natural logarithm has the e as its base, its use is widespread in mathematics and physics. The binary logarithm uses base 2 and is used in computer science. Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations and they were rapidly adopted by navigators, scientists, engineers, and others to perform computations more easily, using slide rules and logarithm tables. The present-day notion of logarithms comes from Leonhard Euler, who connected them to the function in the 18th century. Logarithmic scales reduce wide-ranging quantities to tiny scopes, for example, the decibel is a unit quantifying signal power log-ratios and amplitude log-ratios. In chemistry, pH is a measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and they describe musical intervals, appear in formulas counting prime numbers, inform some models in psychophysics, and can aid in forensic accounting. In the same way as the logarithm reverses exponentiation, the logarithm is the inverse function of the exponential function applied to complex numbers. The discrete logarithm is another variant, it has uses in public-key cryptography, the idea of logarithms is to reverse the operation of exponentiation, that is, raising a number to a power. For example, the power of 2 is 8, because 8 is the product of three factors of 2,23 =2 ×2 ×2 =8. It follows that the logarithm of 8 with respect to base 2 is 3, the third power of some number b is the product of three factors equal to b. More generally, raising b to the power, where n is a natural number, is done by multiplying n factors equal to b. The n-th power of b is written bn, so that b n = b × b × ⋯ × b ⏟ n factors, exponentiation may be extended to by, where b is a positive number and the exponent y is any real number. For example, b−1 is the reciprocal of b, that is, the logarithm of a positive real number x with respect to base b, a positive real number not equal to 1, is the exponent by which b must be raised to yield x
6.
Binary logarithm
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In mathematics, the binary logarithm is the power to which the number 2 must be raised to obtain the value n. That is, for any number x, x = log 2 n ⟺2 x = n. For example, the logarithm of 1 is 0, the binary logarithm of 2 is 1, the binary logarithm of 4 is 2. The binary logarithm is the logarithm to the base 2, the binary logarithm function is the inverse function of the power of two function. As well as log2, alternative notations for the binary logarithm include lg, ld, lb, and log. Binary logarithms can be used to calculate the length of the representation of a number in the numeral system. In computer science, they count the number of steps needed for binary search, other areas in which the binary logarithm is frequently used include combinatorics, bioinformatics, the design of sports tournaments, and photography. Binary logarithms are included in the standard C mathematical functions and other software packages. The integer part of a binary logarithm can be using the find first set operation on an integer value. The fractional part of the logarithm can be calculated efficiently, the powers of two have been known since antiquity, for instance they appear in Euclids Elements, Props. And the binary logarithm of a power of two is just its position in the sequence of powers of two. On this basis, Michael Stifel has been credited with publishing the first known table of binary logarithms in 1544 and his book Arthmetica Integra contains several tables that show the integers with their corresponding powers of two. Reversing the rows of these allow them to be interpreted as tables of binary logarithms. Earlier than Stifel, the 8th century Jain mathematician Virasena is credited with a precursor to the binary logarithm, virasenas concept of ardhacheda has been defined as the number of times a given number can be divided evenly by two. This definition gives rise to a function that coincides with the logarithm on the powers of two, but it is different for other integers, giving the 2-adic order rather than the logarithm. The modern form of a logarithm, applying to any number was considered explicitly by Leonhard Euler in 1739. Euler established the application of binary logarithms to music theory, long before their more significant applications in information theory, as part of his work in this area, Euler published a table of binary logarithms of the integers from 1 to 8, to seven decimal digits of accuracy. Alternatively, it may be defined as ln n/ln 2, where ln is the natural logarithm, using the complex logarithm in this definition allows the binary logarithm to be extended to the complex numbers
7.
International standard
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International standards are standards developed by international standards organizations. International standards are available for consideration and use worldwide, the most prominent organization is the International Organization for Standardization. International standards may be used either by application or by a process of modifying an international standard to suit local conditions. Technical barriers arise when different groups together, each with a large user base. Establishing international standards is one way of preventing or overcoming this problem, the implementation of standards in industry and commerce became highly important with the onset of the Industrial Revolution and the need for high-precision machine tools and interchangeable parts. Henry Maudslay developed the first industrially practical screw-cutting lathe in 1800, maudslays work, as well as the contributions of other engineers, accomplished a modest amount of industry standardization, some companies in-house standards spread a bit within their industries. Joseph Whitworths screw thread measurements were adopted as the first national standard by companies around the country in 1841 and it came to be known as the British Standard Whitworth, and was widely adopted in other countries. By the end of the 19th century differences in standards between companies were making trade increasingly difficult and strained, the Engineering Standards Committee was established in London in 1901 as the worlds first national standards body. After the First World War, similar national bodies were established in other countries, by the mid to late 19th century, efforts were being made to standardize electrical measurement. An important figure was R. E. B, Crompton, who became concerned by the large range of different standards and systems used by electrical engineering companies and scientists in the early 20th century. Many companies had entered the market in the 1890s and all chose their own settings for voltage, frequency, current, adjacent buildings would have totally incompatible electrical systems simply because they had been fitted out by different companies. Crompton could see the lack of efficiency in this system and began to consider proposals for a standard for electric engineering. In 1904, Crompton represented Britain at the Louisiana Purchase Exposition in Saint Louis as part of a delegation by the Institute of Electrical Engineers. He presented a paper on standardisation, which was so well received that he was asked to look into the formation of a commission to oversee the process. By 1906 his work was complete and he drew up a permanent constitution for the first international standards organization, the body held its first meeting that year in London, with representatives from 14 countries. In honour of his contribution to electrical standardisation, Lord Kelvin was elected as the bodys first President, the International Federation of the National Standardizing Associations was founded in 1926 with a broader remit to enhance international cooperation for all technical standards and specifications. The body was suspended in 1942 during World War II, after the war, ISA was approached by the recently formed United Nations Standards Coordinating Committee with a proposal to form a new global standards body. List of international common standards List of technical standard organisations Global Frameworks and standards organized along function lines, accessed 2014 ^ Cordova
8.
Complex number
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A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1. In this expression, a is the part and b is the imaginary part of the complex number. If z = a + b i, then ℜ z = a, ℑ z = b, Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point in the complex plane, a complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the numbers are a field extension of the ordinary real numbers. As well as their use within mathematics, complex numbers have applications in many fields, including physics, chemistry, biology, economics, electrical engineering. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers and he called them fictitious during his attempts to find solutions to cubic equations in the 16th century. Complex numbers allow solutions to equations that have no solutions in real numbers. For example, the equation 2 = −9 has no real solution, Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the unit i where i2 = −1. According to the theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. A complex number is a number of the form a + bi, for example, −3.5 + 2i is a complex number. The real number a is called the part of the complex number a + bi. By this convention the imaginary part does not include the unit, hence b. The real part of a number z is denoted by Re or ℜ. For example, Re = −3.5 Im =2, hence, in terms of its real and imaginary parts, a complex number z is equal to Re + Im ⋅ i. This expression is known as the Cartesian form of z. A real number a can be regarded as a number a + 0i whose imaginary part is 0
9.
International Organization for Standardization
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The International Organization for Standardization is an international standard-setting body composed of representatives from various national standards organizations. Founded on 23 February 1947, the organization promotes worldwide proprietary and it is headquartered in Geneva, Switzerland, and as of March 2017 works in 162 countries. It was one of the first organizations granted general consultative status with the United Nations Economic, ISO, the International Organization for Standardization, is an independent, non-governmental organization, the members of which are the standards organizations of the 162 member countries. It is the worlds largest developer of international standards and facilitates world trade by providing common standards between nations. Nearly twenty thousand standards have been set covering everything from manufactured products and technology to food safety, use of the standards aids in the creation of products and services that are safe, reliable and of good quality. The standards help businesses increase productivity while minimizing errors and waste, by enabling products from different markets to be directly compared, they facilitate companies in entering new markets and assist in the development of global trade on a fair basis. The standards also serve to safeguard consumers and the end-users of products and services, the three official languages of the ISO are English, French, and Russian. The name of the organization in French is Organisation internationale de normalisation, according to the ISO, as its name in different languages would have different abbreviations, the organization adopted ISO as its abbreviated name in reference to the Greek word isos. However, during the meetings of the new organization, this Greek word was not invoked. Both the name ISO and the logo are registered trademarks, the organization today known as ISO began in 1926 as the International Federation of the National Standardizing Associations. ISO is an organization whose members are recognized authorities on standards. Members meet annually at a General Assembly to discuss ISOs strategic objectives, the organization is coordinated by a Central Secretariat based in Geneva. A Council with a membership of 20 member bodies provides guidance and governance. The Technical Management Board is responsible for over 250 technical committees, ISO has formed joint committees with the International Electrotechnical Commission to develop standards and terminology in the areas of electrical and electronic related technologies. Information technology ISO/IEC Joint Technical Committee 1 was created in 1987 to evelop, maintain, ISO has three membership categories, Member bodies are national bodies considered the most representative standards body in each country. These are the members of ISO that have voting rights. Correspondent members are countries that do not have their own standards organization and these members are informed about ISOs work, but do not participate in standards promulgation. Subscriber members are countries with small economies and they pay reduced membership fees, but can follow the development of standards
10.
Exponential function
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In mathematics, an exponential function is a function of the form in which the input variable x occurs as an exponent. A function of the form f = b x + c, as functions of a real variable, exponential functions are uniquely characterized by the fact that the growth rate of such a function is directly proportional to the value of the function. The constant of proportionality of this relationship is the logarithm of the base b. The argument of the function can be any real or complex number or even an entirely different kind of mathematical object. Its ubiquitous occurrence in pure and applied mathematics has led mathematician W. Rudin to opine that the function is the most important function in mathematics. In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same change in the dependent variable. The graph of y = e x is upward-sloping, and increases faster as x increases, the graph always lies above the x -axis but can get arbitrarily close to it for negative x, thus, the x -axis is a horizontal asymptote. The slope of the tangent to the graph at each point is equal to its y -coordinate at that point, as implied by its derivative function. Its inverse function is the logarithm, denoted log, ln, or log e, because of this. The exponential function exp, C → C can be characterized in a variety of equivalent ways, the constant e is then defined as e = exp = ∑ k =0 ∞. The exponential function arises whenever a quantity grows or decays at a proportional to its current value. One such situation is continuously compounded interest, and in fact it was this observation that led Jacob Bernoulli in 1683 to the number lim n → ∞ n now known as e, later, in 1697, Johann Bernoulli studied the calculus of the exponential function. If instead interest is compounded daily, this becomes 365, letting the number of time intervals per year grow without bound leads to the limit definition of the exponential function, exp = lim n → ∞ n first given by Euler. This is one of a number of characterizations of the exponential function, from any of these definitions it can be shown that the exponential function obeys the basic exponentiation identity, exp = exp ⋅ exp which is why it can be written as ex. The derivative of the function is the exponential function itself. More generally, a function with a rate of change proportional to the function itself is expressible in terms of the exponential function and this function property leads to exponential growth and exponential decay. The exponential function extends to a function on the complex plane. Eulers formula relates its values at purely imaginary arguments to trigonometric functions, the exponential function also has analogues for which the argument is a matrix, or even an element of a Banach algebra or a Lie algebra
11.
Natural number
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In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers, the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1,10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory
12.
Natural logarithm
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The natural logarithm of a number is its logarithm to the base of the mathematical constant e, where e is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is written as ln x, loge x, or sometimes, if the base e is implicit. Parentheses are sometimes added for clarity, giving ln, loge or log and this is done in particular when the argument to the logarithm is not a single symbol, to prevent ambiguity. The natural logarithm of x is the power to which e would have to be raised to equal x. The natural log of e itself, ln, is 1, because e1 = e, while the natural logarithm of 1, ln, is 0, since e0 =1. The natural logarithm can be defined for any real number a as the area under the curve y = 1/x from 1 to a. The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, like all logarithms, the natural logarithm maps multiplication into addition, ln = ln + ln . However, logarithms in other bases differ only by a constant multiplier from the natural logarithm, for instance, the binary logarithm is the natural logarithm divided by ln, the natural logarithm of 2. Logarithms are useful for solving equations in which the unknown appears as the exponent of some other quantity, for example, logarithms are used to solve for the half-life, decay constant, or unknown time in exponential decay problems. They are important in many branches of mathematics and the sciences and are used in finance to solve problems involving compound interest, by Lindemann–Weierstrass theorem, the natural logarithm of any positive algebraic number other than 1 is a transcendental number. The concept of the natural logarithm was worked out by Gregoire de Saint-Vincent and their work involved quadrature of the hyperbola xy =1 by determination of the area of hyperbolic sectors. Their solution generated the requisite hyperbolic logarithm function having properties now associated with the natural logarithm, the notations ln x and loge x both refer unambiguously to the natural logarithm of x. log x without an explicit base may also refer to the natural logarithm. This usage is common in mathematics and some scientific contexts as well as in many programming languages, in some other contexts, however, log x can be used to denote the common logarithm. Historically, the notations l. and l were in use at least since the 1730s, finally, in the twentieth century, the notations Log and logh are attested. The graph of the logarithm function shown earlier on the right side of the page enables one to glean some of the basic characteristics that logarithms to any base have in common. Chief among them are, the logarithm of the one is zero. What makes natural logarithms unique is to be found at the point where all logarithms are zero. At that specific point the slope of the curve of the graph of the logarithm is also precisely one