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82-508: The lake formed following the collapse of a slope due to the earthquake of January 23, 1838 at 18:45, measuring 6.9 magnitude on the Richter scale, VIII intensity. The landslide blocked the course of the Bicaz River and the lake formed behind this dam. According to measurements in 1987, the lake has a perimeter of 2,830 m (9,280 ft), and covers an area of 11.4676 ha (28.337 acres);

164-473: A basic tool for measurement and computation in many areas of science and engineering; in these contexts log  x still often means the base ten logarithm. In mathematics log  x usually means to the natural logarithm (base e ). In computer science and information theory, log often refers to binary logarithms (base 2). The following table lists common notations for logarithms to these bases. The "ISO notation" column lists designations suggested by

246-411: A distance of 100 km (62 mi)) a maximum amplitude of 1 micron (1 μm, or 0.001 millimeters) on a seismogram recorded by a Wood-Anderson torsion seismometer. Finally, Richter calculated a table of distance corrections, in that for distances less than 200 kilometers the attenuation is strongly affected by the structure and properties of the regional geology. When Richter presented

328-466: A few) and can vary widely. Millions of minor earthquakes occur every year worldwide, equating to hundreds every hour every day. On the other hand, earthquakes of magnitude ≥8.0 occur about once a year, on average. The largest recorded earthquake was the Great Chilean earthquake of May 22, 1960, which had a magnitude of 9.5 on the moment magnitude scale . Seismologist Susan Hough has suggested that

410-440: A future event because intensity and ground effects depend not only on the magnitude but also on (1) the distance to the epicenter, (2) the depth of the earthquake's focus beneath the epicenter, (3) the location of the epicenter, and (4) geological conditions . ( Based on U.S. Geological Survey documents. ) The intensity and death toll depend on several factors (earthquake depth, epicenter location, and population density, to name

492-447: A great aid to calculations before the invention of computers. Given a positive real number b such that b ≠ 1 , the logarithm of a positive real number x with respect to base  b is the exponent by which b must be raised to yield x . In other words, the logarithm of x to base  b is the unique real number  y such that b y = x {\displaystyle b^{y}=x} . The logarithm

574-682: A magnitude 10 quake may represent a very approximate upper limit for what the Earth's tectonic zones are capable of, which would be the result of the largest known continuous belt of faults rupturing together (along the Pacific coast of the Americas). A research at the Tohoku University in Japan found that a magnitude 10 earthquake was theoretically possible if a combined 3,000 kilometres (1,900 mi) of faults from

656-464: A magnitude of zero to be around the limit of human perceptibility. Third, he specified the Wood–Anderson seismograph as the standard instrument for producing seismograms. Magnitude was then defined as "the logarithm of the maximum trace amplitude, expressed in microns ", measured at a distance of 100 km (62 mi). The scale was calibrated by defining a magnitude 0 shock as one that produces (at

738-446: A product is the sum of the logarithms of the numbers being multiplied; the logarithm of the ratio of two numbers is the difference of the logarithms. The logarithm of the p -th power of a number is p  times the logarithm of the number itself; the logarithm of a p -th root is the logarithm of the number divided by p . The following table lists these identities with examples. Each of the identities can be derived after substitution of

820-618: A wider area, depending on the local geology.) In 1883, John Milne surmised that the shaking of large earthquakes might generate waves detectable around the globe, and in 1899 E. Von Rehbur Paschvitz observed in Germany seismic waves attributable to an earthquake in Tokyo . In the 1920s, Harry O. Wood and John A. Anderson developed the Wood–Anderson seismograph , one of the first practical instruments for recording seismic waves. Wood then built, under

902-507: A yardstick to measure the extent of the event. The resulting effective upper limit of measurement for M L   is about 7 and about 8.5 for M s  . New techniques to avoid the saturation problem and to measure magnitudes rapidly for very large earthquakes are being developed. One of these is based on the long-period P-wave; The other is based on a recently discovered channel wave. The energy release of an earthquake, which closely correlates to its destructive power, scales with

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984-533: Is log b   y . Roughly, a continuous function is differentiable if its graph has no sharp "corners". Moreover, as the derivative of f ( x ) evaluates to ln( b ) b by the properties of the exponential function , the chain rule implies that the derivative of log b   x is given by d d x log b ⁡ x = 1 x ln ⁡ b . {\displaystyle {\frac {d}{dx}}\log _{b}x={\frac {1}{x\ln b}}.} That is,

1066-604: Is a positive real number . (If b is not a positive real number, both exponentiation and logarithm can be defined but may take several values, which makes definitions much more complicated.) One of the main historical motivations of introducing logarithms is the formula log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y,} by which tables of logarithms allow multiplication and division to be reduced to addition and subtraction,

1148-426: Is called the base- b logarithm function or logarithmic function (or just logarithm ). The function log b   x can also be essentially characterized by the product formula log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y . {\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y.} More precisely,

1230-539: Is denoted " log b   x " (pronounced as "the logarithm of x to base  b ", "the base- b logarithm of x ", or most commonly "the log, base  b , of x "). An equivalent and more succinct definition is that the function log b is the inverse function to the function x ↦ b x {\displaystyle x\mapsto b^{x}} . Several important formulas, sometimes called logarithmic identities or logarithmic laws , relate logarithms to one another. The logarithm of

1312-518: Is denoted as log b  ( x ) , or without parentheses, log b   x . When the base is clear from the context or is irrelevant it is sometimes written log  x . The logarithm base 10 is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number  e ≈ 2.718 as its base; its use is widespread in mathematics and physics because of its very simple derivative . The binary logarithm uses base 2 and

1394-464: Is determined from the logarithm of the amplitude of waves recorded by seismographs. Adjustments are included to compensate for the variation in the distance between the various seismographs and the epicenter of the earthquake. The original formula is: where A is the maximum excursion of the Wood-Anderson seismograph , the empirical function A 0 depends only on the epicentral distance of

1476-402: Is essential that the lake was formed by moving the clay mass deposited during the last ice age on the north-western slope of Mount Ghilcoș. Soon after the valley had been closed, the fir forest was flooded , and the trees were petrified, giving a rare peculiarity to the whole landscape. In the first years of forming, the lake has expanded further - about a kilometer higher in the valley of

1558-491: Is exactly one real number x such that b x = y {\displaystyle b^{x}=y} . We let log b : R > 0 → R {\displaystyle \log _{b}\colon \mathbb {R} _{>0}\to \mathbb {R} } denote the inverse of f . That is, log b   y is the unique real number x such that b x = y {\displaystyle b^{x}=y} . This function

1640-405: Is frequently used in computer science . 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

1722-432: Is greater than one. In that case, log b ( x ) is an increasing function . For b < 1 , log b  ( x ) tends to minus infinity instead. When x approaches zero, log b   x goes to minus infinity for b > 1 (plus infinity for b < 1 , respectively). Analytic properties of functions pass to their inverses. Thus, as f ( x ) = b is a continuous and differentiable function , so

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1804-575: Is possible because the logarithm of a product is the sum of the logarithms of the factors: log b ⁡ ( x y ) = log b ⁡ x + log b ⁡ y , {\displaystyle \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 present-day notion of logarithms comes from Leonhard Euler , who connected them to

1886-403: Is related to the number of decimal digits of a positive integer x : The number of digits is the smallest integer strictly bigger than log 10  ( x ) . For example, log 10 (5986) is approximately 3.78 . The next integer above it is 4, which is the number of digits of 5986. Both the natural logarithm and the binary logarithm are used in information theory , corresponding to

1968-411: Is written as f ( x ) = b . When b is positive and unequal to 1, we show below that f is invertible when considered as a function from the reals to the positive reals. Let b be a positive real number not equal to 1 and let f ( x ) = b . It is a standard result in real analysis that any continuous strictly monotonic function is bijective between its domain and range. This fact follows from

2050-548: The 3 ⁄ 2 power of the shaking amplitude (see Moment magnitude scale for an explanation). Thus, a difference in magnitude of 1.0 is equivalent to a factor of 31.6 ( = ( 10 1.0 ) ( 3 / 2 ) {\displaystyle =({10^{1.0}})^{(3/2)}} ) in the energy released; a difference in magnitude of 2.0 is equivalent to a factor of 1000 ( = ( 10 2.0 ) ( 3 / 2 ) {\displaystyle =({10^{2.0}})^{(3/2)}} ) in

2132-633: The International Organization for Standardization . The history of logarithms in seventeenth-century Europe saw the discovery of a new function that extended the realm of analysis beyond the scope of algebraic methods. The method of logarithms was publicly propounded by John Napier in 1614, in a book titled Mirifici Logarithmorum Canonis Descriptio ( Description of the Wonderful Canon of Logarithms ). Prior to Napier's invention, there had been other techniques of similar scopes, such as

2214-529: The Japan Trench to the Kuril–Kamchatka Trench ruptured together and moved by 60 metres (200 ft) (or if a similar large-scale rupture occurred elsewhere). Such an earthquake would cause ground motions for up to an hour, with tsunamis hitting shores while the ground is still shaking, and if this kind of earthquake occurred, it would probably be a 1-in-10,000-year event. Prior to the development of

2296-439: The acidity of an aqueous solution . Logarithms are commonplace in scientific formulae , and in measurements of the complexity of algorithms and of geometric objects called fractals . They help to describe frequency ratios of musical intervals , appear in formulas counting prime numbers or approximating factorials , inform some models in psychophysics , and can aid in forensic accounting . The concept of logarithm as

2378-431: The decimal number system: log 10 ( 10 x )   = log 10 ⁡ 10   + log 10 ⁡ x   =   1 + log 10 ⁡ x . {\displaystyle \log _{10}\,(\,10\,x\,)\ =\;\log _{10}10\ +\;\log _{10}x\ =\ 1\,+\,\log _{10}x\,.} Thus, log 10  ( x )

2460-413: The exponential function in the 18th century, and who also introduced the letter e as the base of natural logarithms. Logarithmic scales reduce wide-ranging quantities to smaller scopes. For example, the decibel (dB) is a unit used to express ratio as logarithms , mostly for signal power and amplitude (of which sound pressure is a common example). In chemistry, pH is a logarithmic measure for

2542-501: The function now known as the natural logarithm began as an attempt to perform a quadrature of a rectangular hyperbola by Grégoire de Saint-Vincent , a Belgian Jesuit residing in Prague. Archimedes had written The Quadrature of the Parabola in the third century BC, but a quadrature for the hyperbola eluded all efforts until Saint-Vincent published his results in 1647. The relation that

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2624-579: The intermediate value theorem . Now, f is strictly increasing (for b > 1 ), or strictly decreasing (for 0 < b < 1 ), is continuous, has domain R {\displaystyle \mathbb {R} } , and has range R > 0 {\displaystyle \mathbb {R} _{>0}} . Therefore, f is a bijection from R {\displaystyle \mathbb {R} } to R > 0 {\displaystyle \mathbb {R} _{>0}} . In other words, for each positive real number y , there

2706-421: The logarithm to base b is the inverse function of exponentiation with base b . That means that the logarithm of a number  x to the base   b is the exponent to which b must be raised to produce x . For example, since 1000 = 10 , the logarithm base   10 {\displaystyle 10} of 1000 is 3 , or log 10  (1000) = 3 . The logarithm of x to base   b

2788-459: The logarithmic character of the original and are scaled to have roughly comparable numeric values (typically in the middle of the scale). Due to the variance in earthquakes, it is essential to understand the Richter scale uses common logarithms simply to make the measurements manageable (i.e., a magnitude 3 quake factors 10³ while a magnitude 5 quake factors 10 and has seismometer readings 100 times larger). The Richter magnitude of an earthquake

2870-510: The prosthaphaeresis or the use of tables of progressions, extensively developed by Jost Bürgi around 1600. Napier coined the term for logarithm in Middle Latin, logarithmus , literally meaning ' ratio-number ' , derived from the Greek logos ' proportion, ratio, word ' + arithmos ' number ' . The common logarithm of a number is the index of that power of ten which equals

2952-407: The slope of the tangent touching the graph of the base- b logarithm at the point ( x , log b  ( x )) equals 1/( x  ln( b )) . The derivative of ln( x ) is 1/ x ; this implies that ln( x ) is the unique antiderivative of 1/ x that has the value 0 for x = 1 . It is this very simple formula that motivated to qualify as "natural" the natural logarithm; this is also one of

3034-467: The surface-wave magnitude (M S ) and body wave magnitude (M B ) scales. The Richter scale was defined in 1935 for particular circumstances and instruments; the particular circumstances refer to it being defined for Southern California and "implicitly incorporates the attenuative properties of Southern California crust and mantle." The particular instrument used would become saturated by strong earthquakes and unable to record high values. The scale

3116-400: The x - and the y -coordinates (or upon reflection at the diagonal line x = y ), as shown at the right: a point ( t , u = b ) on the graph of f yields a point ( u , t = log b   u ) on the graph of the logarithm and vice versa. As a consequence, log b  ( x ) diverges to infinity (gets bigger than any given number) if x grows to infinity, provided that b

3198-576: The "Richter scale", , especially the local magnitude M L   and the surface wave M s   scale. In addition, the body wave magnitude , mb , and the moment magnitude , M w  , abbreviated MMS, have been widely used for decades. A couple of new techniques to measure magnitude are in the development stage by seismologists. All magnitude scales have been designed to give numerically similar results. This goal has been achieved well for M L  , M s  , and M w  . The mb  scale gives somewhat different values than

3280-451: The "magnitude scale". This was later revised and renamed the local magnitude scale , denoted as ML or M L  . Because of various shortcomings of the original M L   scale, most seismological authorities now use other similar scales such as the moment magnitude scale (M w  ) to report earthquake magnitudes, but much of the news media still erroneously refers to these as "Richter" magnitudes. All magnitude scales retain

3362-614: The 1900s, it has been the recreational spa tourism that has brought development to the tourist services of this area. Richter magnitude scale The Richter scale ( / ˈ r ɪ k t ər / ), also called the Richter magnitude scale , Richter's magnitude scale , and the Gutenberg–Richter scale , is a measure of the strength of earthquakes , developed by Charles Richter in collaboration with Beno Gutenberg , and presented in Richter's landmark 1935 paper, where he called it

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3444-407: The 1970s, because it allows, at the expense of precision, much faster computation than techniques based on tables. A deeper study of logarithms requires the concept of a function . A function is a rule that, given one number, produces another number. An example is the function producing the x -th power of b from any real number  x , where the base  b is a fixed number. This function

3526-451: The M s   scale. A spectral analysis is required to obtain M 0  . In contrast, the other magnitudes are derived from a simple measurement of the amplitude of a precisely defined wave. All scales, except M w  , saturate for large earthquakes, meaning they are based on the amplitudes of waves that have a wavelength shorter than the rupture length of the earthquakes. These short waves (high-frequency waves) are too short

3608-407: The advance of science, especially astronomy . They were critical to advances in surveying , celestial navigation , and other domains. Pierre-Simon Laplace called logarithms As the function f ( x ) = b is the inverse function of log b   x , it has been called an antilogarithm . Nowadays, this function is more commonly called an exponential function . A key tool that enabled

3690-457: The amount of energy released, and each increase of 0.2 corresponds to approximately a doubling of the energy released. Events with magnitudes greater than 4.5 are strong enough to be recorded by a seismograph anywhere in the world, so long as its sensors are not located in the earthquake's shadow . The following describes the typical effects of earthquakes of various magnitudes near the epicenter. The values are typical and may not be exact in

3772-529: The auspices of the California Institute of Technology and the Carnegie Institute , a network of seismographs stretching across Southern California . He also recruited the young and unknown Charles Richter to measure the seismograms and locate the earthquakes generating the seismic waves. In 1931, Kiyoo Wadati showed how he had measured, for several strong earthquakes in Japan, the amplitude of

3854-451: The base is given by: b = x 1 y , {\displaystyle b=x^{\frac {1}{y}},} which can be seen from taking the defining equation x = b log b ⁡ x = b y {\displaystyle x=b^{\,\log _{b}x}=b^{y}} to the power of 1 y . {\displaystyle {\tfrac {1}{y}}.} Among all choices for

3936-405: The base, three are particularly common. These are b = 10 , b = e (the irrational mathematical constant e ≈ 2.71828183 ), and b = 2 (the binary logarithm ). In mathematical analysis , the logarithm base e is widespread because of analytical properties explained below. On the other hand, base 10 logarithms (the common logarithm ) are easy to use for manual calculations in

4018-453: The common logarithms of trigonometric functions . Another critical application was the slide rule , a pair of logarithmically divided scales used for calculation. The non-sliding logarithmic scale, Gunter's rule , was invented shortly after Napier's invention. William Oughtred enhanced it to create the slide rule—a pair of logarithmic scales movable with respect to each other. Numbers are placed on sliding scales at distances proportional to

4100-638: The conditions and time of the lake formation are very much discussed. During the forming, the lake area was a hardly accessible area, economically unexplored. According to Franz Herbich, the Red Lake was formed in 1838. This is also justified by the earthquake of January 23, 1838, which was repeated in February and could have caused a landslide . Another year of forming is 1837, which can be argued by very violent storms and heavy rains. About this period writes Ditrói Puskás Ferenc in his work, "The History of Borsec ". It

4182-400: The differences between their logarithms. Sliding the upper scale appropriately amounts to mechanically adding logarithms, as illustrated here: For example, adding the distance from 1 to 2 on the lower scale to the distance from 1 to 3 on the upper scale yields a product of 6, which is read off at the lower part. The slide rule was an essential calculating tool for engineers and scientists until

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4264-528: The energy released by an earthquake; another scale, the Mercalli intensity scale , classifies earthquakes by their effects , from detectable by instruments but not noticeable, to catastrophic. The energy and effects are not necessarily strongly correlated; a shallow earthquake in a populated area with soil of certain types can be far more intense in impact than a much more energetic deep earthquake in an isolated area. Several scales have been historically described as

4346-897: The energy released. The elastic energy radiated is best derived from an integration of the radiated spectrum, but an estimate can be based on mb  because most energy is carried by the high-frequency waves. These formulae for Richter magnitude   M L   {\displaystyle \ M_{\mathsf {L}}\ } are alternatives to using Richter correlation tables based on Richter standard seismic event (   M L = 0   , {\displaystyle {\big (}\ M_{\mathsf {L}}=0\ ,}   A = 0.001   m m   , {\displaystyle \ A=0.001\ {\mathsf {mm}}\ ,}   D = 100   k m   )   . {\displaystyle \ D=100\ {\mathsf {km}}\ {\big )}~.} In

4428-433: The following formula: log b ⁡ x = log k ⁡ x log k ⁡ b . {\displaystyle \log _{b}x={\frac {\log _{k}x}{\log _{k}b}}.} Typical scientific calculators calculate the logarithms to bases 10 and e . Logarithms with respect to any base  b can be determined using either of these two logarithms by

4510-659: The formulas below,   Δ   {\displaystyle \ \Delta \ } is the epicentral distance in kilometers , and   Δ ∘   {\displaystyle \ \Delta ^{\circ }\ } is the same distance represented as sea level great circle degrees. The Lillie empirical formula is: Lahr's empirical formula proposal is: and The Bisztricsany empirical formula (1958) for epicentre distances between 4° and 160° is: The Tsumura empirical formula is: The Tsuboi (University of Tokyo) empirical formula is: Logarithm In mathematics ,

4592-482: The inverse of exponentiation extends to other mathematical structures as well. However, in general settings, the logarithm tends to be a multi-valued function. For example, the complex logarithm is the multi-valued inverse of the complex exponential function. Similarly, the discrete logarithm is the multi-valued inverse of the exponential function in finite groups; it has uses in public-key cryptography . Addition , multiplication , and exponentiation are three of

4674-462: The log base 2  ; and in photography rescaled base 2 logarithms are used to measure exposure values , light levels , exposure times , lens apertures , and film speeds in "stops". The abbreviation log  x is often used when the intended base can be inferred based on the context or discipline, or when the base is indeterminate or immaterial. Common logarithms (base 10), historically used in logarithm tables and slide rules, are

4756-433: The logarithm definitions x = b log b ⁡ x {\displaystyle x=b^{\,\log _{b}x}} or y = b log b ⁡ y {\displaystyle y=b^{\,\log _{b}y}} in the left hand sides. The logarithm log b   x can be computed from the logarithms of x and b with respect to an arbitrary base  k using

4838-434: The logarithm provides between a geometric progression in its argument and an arithmetic progression of values, prompted A. A. de Sarasa to make the connection of Saint-Vincent's quadrature and the tradition of logarithms in prosthaphaeresis , leading to the term "hyperbolic logarithm", a synonym for natural logarithm. Soon the new function was appreciated by Christiaan Huygens , and James Gregory . The notation Log y

4920-528: The logarithm to any base b > 1 is the only increasing function f from the positive reals to the reals satisfying f ( b ) = 1 and f ( x y ) = f ( x ) + f ( y ) . {\displaystyle f(xy)=f(x)+f(y).} As discussed above, the function log b is the inverse to the exponential function x ↦ b x {\displaystyle x\mapsto b^{x}} . Therefore, their graphs correspond to each other upon exchanging

5002-926: The lookups of the two logarithms, calculating their sum or difference, and looking up the antilogarithm is much faster than performing the multiplication by earlier methods such as prosthaphaeresis , which relies on trigonometric identities . Calculations of powers and roots are reduced to multiplications or divisions and lookups by c d = ( 10 log 10 ⁡ c ) d = 10 d log 10 ⁡ c {\displaystyle c^{d}=\left(10^{\,\log _{10}c}\right)^{d}=10^{\,d\log _{10}c}} and c d = c 1 d = 10 1 d log 10 ⁡ c . {\displaystyle {\sqrt[{d}]{c}}=c^{\frac {1}{d}}=10^{{\frac {1}{d}}\log _{10}c}.} Trigonometric calculations were facilitated by tables that contained

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5084-478: The magnitude scale, the only measure of an earthquake's strength or "size" was a subjective assessment of the intensity of shaking observed near the epicenter of the earthquake, categorized by various seismic intensity scales such as the Rossi–Forel scale . ("Size" is used in the sense of the quantity of energy released, not the size of the area affected by shaking, though higher-energy earthquakes do tend to affect

5166-1221: The mantissa, as the characteristic can be easily determined by counting digits from the decimal point. The characteristic of 10 · x is one plus the characteristic of x , and their mantissas are the same. Thus using a three-digit log table, the logarithm of 3542 is approximated by log 10 ⁡ 3542 = log 10 ⁡ ( 1000 ⋅ 3.542 ) = 3 + log 10 ⁡ 3.542 ≈ 3 + log 10 ⁡ 3.54 {\displaystyle {\begin{aligned}\log _{10}3542&=\log _{10}(1000\cdot 3.542)\\&=3+\log _{10}3.542\\&\approx 3+\log _{10}3.54\end{aligned}}} Greater accuracy can be obtained by interpolation : log 10 ⁡ 3542 ≈ 3 + log 10 ⁡ 3.54 + 0.2 ( log 10 ⁡ 3.55 − log 10 ⁡ 3.54 ) {\displaystyle \log _{10}3542\approx {}3+\log _{10}3.54+0.2(\log _{10}3.55-\log _{10}3.54)} The value of 10 can be determined by reverse look up in

5248-407: The moment magnitude scale (MMS) is most common, although M s   is also reported frequently. The seismic moment , M 0   , is proportional to the area of the rupture times the average slip that took place in the earthquake, thus it measures the physical size of the event. M w   is derived from it empirically as a quantity without units, just a number designed to conform to

5330-447: The most fundamental arithmetic operations. The inverse of addition is subtraction , and the inverse of multiplication is division . Similarly, a logarithm is the inverse operation of exponentiation . Exponentiation is when a number b , the base , is raised to a certain power y , the exponent , to give a value x ; this is denoted b y = x . {\displaystyle b^{y}=x.} For example, raising 2 to

5412-459: The north-east, and Muntele Ucigaș (The Killer Mount) to the east. The lake is powered by four large streams and 12 temporary water courses, of which the most important are Vereșchiu, Licaș, Suhardul, and Pârâul Oii (Oaia). The Bicaz River streams out of the Red Lake and continues towards the Bicaz Gorge, about 4 km (2.5 mi) to the north-east. Although the Red Lake is a young formation,

5494-425: 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. Some of these methods used tables derived from trigonometric identities. Such methods are called prosthaphaeresis . Invention of

5576-433: The other scales. The reason for so many different ways to measure the same thing is that at different distances, for different hypocentral depths, and for different earthquake sizes, the amplitudes of different types of elastic waves must be measured. M L   is the scale used for the majority of earthquakes reported (tens of thousands) by local and regional seismological observatories. For large earthquakes worldwide,

5658-414: The power of 3 gives 8 : 2 3 = 8. {\displaystyle 2^{3}=8.} The logarithm of base b is the inverse operation, that provides the output y from the input x . That is, y = log b ⁡ x {\displaystyle y=\log _{b}x} is equivalent to x = b y {\displaystyle x=b^{y}} if b

5740-424: The practical use of logarithms was the table of logarithms . The first such table was compiled by Henry Briggs in 1617, immediately after Napier's invention but with the innovation of using 10 as the base. Briggs' first table contained the common logarithms of all integers in the range from 1 to 1000, with a precision of 14 digits. Subsequently, tables with increasing scope were written. These tables listed

5822-475: The previous formula: log b ⁡ x = log 10 ⁡ x log 10 ⁡ b = log e ⁡ x log e ⁡ b . {\displaystyle \log _{b}x={\frac {\log _{10}x}{\log _{10}b}}={\frac {\log _{e}x}{\log _{e}b}}.} Given a number x and its logarithm y = log b   x to an unknown base  b ,

5904-435: The relative magnitudes of different earthquakes. Additional developments were required to produce a practical method of assigning an absolute measure of magnitude. First, to span the wide range of possible values, Richter adopted Gutenberg's suggestion of a logarithmic scale, where each step represents a tenfold increase of magnitude, similar to the magnitude scale used by astronomers for star brightness . Second, he wanted

5986-451: The resulting scale in 1935, he called it (at the suggestion of Harry Wood) simply a "magnitude" scale. "Richter magnitude" appears to have originated when Perry Byerly told the press that the scale was Richter's and "should be referred to as such." In 1956, Gutenberg and Richter, while still referring to "magnitude scale", labelled it "local magnitude", with the symbol M L  , to distinguish it from two other scales they had developed,

6068-1046: The same table, since the logarithm is a monotonic function . The product and quotient of two positive numbers c and d were routinely calculated as the sum and difference of their logarithms. The product  cd or quotient  c / d came from looking up the antilogarithm of the sum or difference, via the same table: c d = 10 log 10 ⁡ c 10 log 10 ⁡ d = 10 log 10 ⁡ c + log 10 ⁡ d {\displaystyle cd=10^{\,\log _{10}c}\,10^{\,\log _{10}d}=10^{\,\log _{10}c\,+\,\log _{10}d}} and c d = c d − 1 = 10 log 10 ⁡ c − log 10 ⁡ d . {\displaystyle {\frac {c}{d}}=cd^{-1}=10^{\,\log _{10}c\,-\,\log _{10}d}.} For manual calculations that demand any appreciable precision, performing

6150-434: The shaking observed at various distances from the epicenter. He then plotted the logarithm of the amplitude against the distance and found a series of curves that showed a rough correlation with the estimated magnitudes of the earthquakes. Richter resolved some difficulties with this method and then, using data collected by his colleague Beno Gutenberg , he produced similar curves, confirming that they could be used to compare

6232-460: The station, δ {\displaystyle \delta } . In practice, readings from all observing stations are averaged after adjustment with station-specific corrections to obtain the M L   value. Because of the logarithmic basis of the scale, each whole number increase in magnitude represents a tenfold increase in measured amplitude. In terms of energy, each whole number increase corresponds to an increase of about 31.6 times

6314-509: The stream, but over time the natural dam eroded, the water level stabilizing at the current level. The surroundings of the lake have a pleasant microclimate . The average multiannual temperature is 8 °C (46 °F), above the 6 °C (43 °F) average of the intramontane depressions. The valley is virtually free of winds , very clean air rich in natural aerosols , scenic surroundings provide excellent conditions for those who are seeking for sources of rapid regeneration naturally. Since

6396-488: The use of nats or bits as the fundamental units of information, respectively. Binary logarithms are also used in computer science , where the binary system is ubiquitous; in music theory , where a pitch ratio of two (the octave ) is ubiquitous and the number of cents between any two pitches is a scaled version of the binary logarithm, or log 2 times 1200, of the pitch ratio (that is, 100 cents per semitone in conventional equal temperament ), or equivalently

6478-455: The values of log 10   x for any number  x in a certain range, at a certain precision. Base-10 logarithms were universally used for computation, hence the name common logarithm, since numbers that differ by factors of 10 have logarithms that differ by integers. The common logarithm of x can be separated into an integer part and a fractional part , known as the characteristic and mantissa . Tables of logarithms need only include

6560-596: The volume of water that accumulates is 587,503 cubic metres (768,425 cu yd). The lake was formed at an altitude of 983 m (3,225 ft), in a depression with a predominant subalpine climate. The Red Lake is located between Suhardul Mic and Suhardul Mare peaks on the north side, the Podu Calului Mountains to the south-west, the Licaș and Chișhovoș Mountains to the north-west, the Făgetul Ciucului peak to

6642-673: Was adopted by Leibniz in 1675, and the next year he connected it to the integral ∫ d y y . {\textstyle \int {\frac {dy}{y}}.} Before Euler developed his modern conception of complex natural logarithms, Roger Cotes had a nearly equivalent result when he showed in 1714 that log ⁡ ( cos ⁡ θ + i sin ⁡ θ ) = i θ . {\displaystyle \log(\cos \theta +i\sin \theta )=i\theta .} By simplifying difficult calculations before calculators and computers became available, logarithms contributed to

6724-437: Was replaced in the 1970s by the moment magnitude scale (MMS, symbol M w  ); for earthquakes adequately measured by the Richter scale, numerical values are approximately the same. Although values measured for earthquakes now are M w  , they are frequently reported by the press as Richter values, even for earthquakes of magnitude over 8, when the Richter scale becomes meaningless. The Richter and MMS scales measure

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