SUMMARY / RELATED TOPICS

In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x. In the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication; the logarithm of x to base b is denoted as logb , or without parentheses, logb x, or without the explicit base, log x, when no confusion is possible, or when the base does not matter such as in big O notation. More exponentiation allows any positive real number as base to be raised to any real power, always producing a positive result, so logb for any two positive real numbers b and x, where b is not equal to 1, is always a unique real number y. More explicitly, the defining relation between exponentiation and logarithm is: log b ⁡ = y if b y = x and x > 0 and b > 0 and b ≠ 1. For example, log2 64 = 6, as 26 = 64; the logarithm base 10 is called the common logarithm and has many applications in science and engineering.

The natural logarithm has the number e as its base. The binary logarithm uses base 2 and is used in computer science. Logarithms are examples of concave functions. Logarithms were introduced by John Napier in 1614 as a means of simplifying calculations, they were adopted by navigators, engineers 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 of the fact—important in its own right—that the logarithm of a product is the sum of the logarithms of the factors: log b ⁡ = log b ⁡ x + log b ⁡ y, provided that b, x and y are all positive and b ≠ 1. The slide rule 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 the exponential function in the 18th century, who introduced the letter e as the base of natural logarithms. Logarithmic scales reduce wide-ranging quantities to tiny scopes.

For example, the decibel is a unit used to express ratio as logarithms for signal power and amplitude. In chemistry, pH is a logarithmic measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, in measurements of the complexity of algorithms and of geometric objects called fractals, they help describing frequency ratios of musical intervals, appear in formulas counting prime numbers or approximating factorials, inform some models in psychophysics, can aid in forensic accounting. In the same way as the logarithm reverses exponentiation, the complex logarithm is the inverse function of the exponential function applied to complex numbers; the modular discrete logarithm is another variant. Addition and exponentiation are three of the most fundamental arithmetic operations. Addition, the simplest of these, is undone by subtraction: when you add 5 to x to get x + 5, to reverse this operation you need to subtract 5 from x + 5. Multiplication, the next-simplest operation, is undone by division: if you multiply x by 5 to get 5x, you must divide 5x by 5 in order to return to the original expression x.

Logarithms undo a fundamental arithmetic operation, exponentiation. Exponentiation is. For example, raising 2 to the power 3 equals 8: 2 3 = 2 × 2 × 2 = 8 The general case is when you raise a number b to the power of y to get x: b y = x The number b is referred to as the base of this expression; the base is the number, raised to a particular power—in the above example, the base of the expression 2 3 = 8 is 2. It is easy to make the base the subject of the expression: all you have to do is take the y-th root of both sides; this gives: b = x y It is less easy to make y the subject of the expression. Logarithms allow us to do this: y = logb xThis expression means that y is equal to the power that you need to raise b to in order to get x; this operation undoes exponentiation because the logarithm of x tells you the exponent that the base has been raised to. This subsection contains a short overview of the exponentiation operation, fundamental to understanding logarithms. Raising b to the n-th 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 ×

Andrew K. Benton is an American lawyer and academic administrator who served as the seventh president of Pepperdine University. Benton was born in Kansas, he graduated from Oklahoma Christian University, where he earned a bachelor's degree in American Studies in 1974, he earned a J. D. degree form Oklahoma City University. Benton began his career as a lawyer in Oklahoma, he was an assistant to the president of his alma mater, Oklahoma Christian University, from 1975 to 1984. Benton joined Pepperdine University in 1984, he served as its vice president from 1991 to 1999. He served as its president from 2000 to 2019, becoming the longest-serving person to hold the office. During the course of his presidency, "Total University assets have increased by more than \$1 billion." His presidency saw the construction of the Drescher graduate campus and the establishment of campuses in Lausanne and Shanghai. When Jim Gash succeeded him in 2019, Benton took on the role of president emeritus. A road on the university's Malibu campus was renamed Benton Way in honor of his service to the university.

While serving at Pepperdine, Benton was named the Vice Chair, the Chair, of the Board of Directors of the National Association of Independent Colleges and Universities. Benton was honored as a distinguished alumnus by Oklahoma Christian University in 2000. With his wife Deborah, Benton has a daughter, he is a member of the Churches of Christ

Merion Estes is a Los Angeles-based painter. She earned a B. F. A. at the University of New Mexico, in Albuquerque, an M. F. A. at the University of Colorado, in Boulder. Estes was raised in San Diego from the age of four, she first showed her work at the Woman's Building in Los Angeles. As a founding member of Grandview 1 & 2, she was involved in the beginnings of Los Angeles feminist art organizations including Womanspace, the feminist arts group, "Double X," along with artists Judy Chicago, Nancy Buchanan, Faith Wilding, Nancy Youdelman. In 2014, Un-Natural, at the Los Angeles Municipal Art Gallery in Los Angeles, which included Estes' work, was named one of the best shows in a non-profit institution in the United States by the International Association of Art Critics. Estes in the 1970s through the 1980s was a pioneer in the Decoration movement. Suzanne Muchnic wrote in the Los Angeles Times, "What's interesting about this art is that Estes pulls warm textures from slick materials and builds soft forms from hard-edge patterns…her real concerns are light and color transformation achieved by repetition and a rigid system."Estes was featured in a five-year solo survey of her work at the Los Angeles Municipal Art Gallery, Barnsdall Park, in 1979, curated by Josine Ianco-Starrels.

A group exhibition of the Double X group was presented the next year at Los Angeles Contemporary Exhibitions. Estes was included in several exhibits of work by artists with studios Downtown L. A. including the first exhibition at the ARCO Center Gallery in 1976. In 2005, Fisher Galleries at the University of Southern California organized Contemporary Soliloquies on the Natural World: Karen Carson, Merion Estes, Constance Mallinson, Margaret Nielsen, Takako Yamaguchi; the exhibition was curated by Max Schultz. Los Angeles Times art critic David Pagel wrote that Estes' paintings were a "dizzying collision of extravagantly patterned fabrics onto which the artist has splashed and stained various mixtures of oil and acrylic." He cited the "funky verve of her collaged paintings, which are the show's high point." In the exhibition catalogue, curator Schultz writes that Estes' art has "rooted in the nether and cloudborn worlds of sea and sky intensely for enough years to produce a complex artful weave of realistic and abstract cellular, animal and mineral forms."In September 2006, Pomona College, California, mounted a major 35-year retrospective of Estes' work.

Michael Duncan, in Art in America wrote, Estes is "one of L. A.'s most underrated, yet most inventive artists who has explored the intersection of nature and decoration in brash, vigorously constructed, brightly colored oil and acrylic paintings."Critic Betty Brown wrote a catalogue essay for Lost Horizons, at Galerie Anais, Santa Monica, describing Estes' paintings at the exhibition in 2009, as both beautiful and difficult. "They maintain this apparent contradictory state because the joy we feel through sight--the sheer visual delight derived from her unabashedly exuberant shapes and colors and textures--is tempered by the sorrow we feel as we recognize the environmental devastation undermining the luxurious abundance of her scintillating surfaces."Estes' work is included in an online exhibition of work by artists working with the theme of nature. Curator and writer Constance Mallinson writes, "Through her multiple references to natural life from the sea to the air, Estes evokes a sublime sense of endangered and fragile beauty that extends globally."About Un-natural, an exhibition sponsored by the Los Angeles Municipal Art Gallery, installed at Barnsdall Park, Fabrik magazine art critic Peter Frank writes that the "expansive and complex formulations of Merion Estes, brimming with stylized references and visual montages…seem to be coding and recording how humanity interacts with nature."

The exhibition earned honors from the International Association of Art Critics. Dystopia, CB1 Gallery, Los Angeles, 2015 Un-Natural, Municipal Art Gallery at Barnsall Park, Los Angeles, 2012 Painting Per Se: Los Angeles Paintings from the Seventies, David Richard Contemporary Gallery, Santa Fe, New Mexico, 2011 Urbanature, An online exhibition of artists working with the theme of nature, Times Quotidian, Constance Mallison, curator, 2010 Merion Estes: Lost Horizons, Galerie Anais, Bergamot Station, Santa Monica, California, 2009 Women Artists in So Cal: Then & Now, Track 16 Gallery, Bergamot Station, Santa Monica, California, 2007 Merion Estes: Recent Paintings, Cardwell-Jimmerson Gallery, Culver City, California, 2007 A Sea of Possibilities: Works by Merion Estes 1971-2006, Pomona College Museum of Art, California, 2006 Contemporary Soliloquies on the Natural World: Karen Carson, Merion Estes, Constance Mallinson, Margaret Nielsen, Takako Yamaguchi, USC Fisher Galleries, Los Angeles, 2005 LAPD, Rosamond Felsen Gallery, Bergamot Station, Santa Monica, California, 2003 Post Cool, San Jose Museum of Art, San Jose, California, 2004 Joan Mitchell Foundation Grant, Residency: Santa Fe Art Institute, Santa Fe, New Mexico, 2007 J. Paul Getty Trust Fund for the Visual Arts, Artist Fellowship, 1996 CETA Grant, Art in Public Spaces, 1980 California Arts Commission grant, 1980 CB1 Gallery https://www.youtube.com/watch?v=L40KeWsQFgo

The 2008 Italian Grand Prix was a Formula One motor race held on 14 September 2008 at the Autodromo Nazionale Monza, Italy. It was the 14th race of the 2008 Formula One season; the 53-lap race was won by Sebastian Vettel for the Toro Rosso team after starting from pole position. Heikki Kovalainen finished second in a McLaren, Robert Kubica third in a BMW Sauber. Vettel began the race, started under the safety car, ahead of Kovalainen in second. Red Bull's Mark Webber started from third. Rain early in the race allowed Vettel to establish a solid lead over Kovalainen, which he extended as the track dried. Kubica and Fernando Alonso finished in the top four after starting from 11th and eighth, respectively. McLaren driver and Drivers' Championship leader Lewis Hamilton was able to move through the field after qualifying in 15th, finishing in seventh, one place behind rival Felipe Massa, of Ferrari. Vettel's victory made him the youngest driver to win a Formula One race, at 21 years 73 days in addition to giving Toro Rosso its maiden Formula One win despite using a 2007-spec engine.

Vettel's record was broken by Max Verstappen aged 18 years and 228 days at the 2016 Spanish Grand Prix. Massa scored one point more than Hamilton, narrowing the McLaren driver's lead in the Championship once more with four races remaining. However, Kovalainen's second-placed finish put McLaren closer to catching Ferrari in the Constructors' Championship. Heading into the 14th race of the season, McLaren driver Lewis Hamilton led the Drivers' Championship with 76 points. Behind Hamilton and Massa in the Drivers' Championship, Robert Kubica was third on 58 points in a BMW Sauber, Massa's Ferrari teammate Kimi Räikkönen fourth on 57 points. Kubica's teammate Nick Heidfeld was fifth on 49 points. In the Constructors' Championship, Ferrari were leading on 131 points and McLaren were second on 119 points, 12 points behind. BMW Sauber were third with 107 points. In the battle for fourth-place Toyota had 41 points, ahead of Renault with 36 points. After the previous race, Hamilton was penalised for cutting a chicane and gaining an advantage over Räikkönen in the closing laps.

This meant. While McLaren had lodged an appeal against the decision, the result would not be heard until September 23. Reflecting on the controversy, Norbert Haug, Vice President of Mercedes-Benz Motorsport, said: Our disappointment was big, when the stewards took away victory from him and the team. However, we are fighters. If we would have needed a better motivation for the last five races of the season we have it now; when we went to the airport last Sunday evening, Lewis said to me - preferably we now want to win all remaining races, don't we? I have no objection. Following the rain-soaked Belgian Grand Prix, Massa expressed his hope that the Italian Grand Prix would be held on a dry track, saying "it would be nice not to have the rain here and not have any opportunity to have a consistent race." The close nature of the Championship meant that Ferrari's home race had the potential to be a turning point in the season. Testing at the Monza circuit in early September had indicated that the performance gap between McLaren and Ferrari had narrowed.

Despite this, Ferrari were hoping that their straight-line speed would be better suited to Monza's long straights. Ferrari confirmed on September 12 that Räikkönen would continue to drive for the team until at least 2010, ending media speculation that he could be about to retire. Three practice sessions were held before the race: the first was held on Friday morning and the second on Friday afternoon. Both sessions lasted 90 minutes; the third session was lasted an hour. All three sessions were affected by rain, with only the second giving teams any chance to run dry-weather tyres. Adrian Sutil was quickest with a time of 1:32.842 in the first session, more than half a second quicker than the next fastest drivers Rubens Barrichello and Giancarlo Fisichella. Timo Glock was the next quickest four seconds behind Sutil. Conditions were so poor; the session was stopped four minutes early, though by that time no cars had entered the track for at least ten minutes. The track dried out in the second session.

Kubica was the next fastest, with his BMW teammate Heidfeld in third. Hamilton, Nico Rosberg and Massa rounded out the top six. Glock was fastest in the final session with a time of 1:35.464, followed by Sebastian Vettel, Jarno Trulli, Kazuki Nakajima and Heidfeld. Hamilton only completed eight laps, finished last with a time 11 seconds behind Glock; the qualifying session on Saturday afternoon was split into three parts. The first part ran for 20 minutes and eliminated the cars from qualifying that finished the session 16th or lower; the second part of qualifying lasted 15 minutes and eliminated cars that finished in positions 11 to 15. The final part of qualifying determined the positions from first to tenth, decided pole position. Cars which competed in the final session of qualifying were not allowed to refuel before the race, as such carried more fuel than in the previous sessions. Vettel became the youngest driver in the history of Formula One to take pole position, with a time of 1:37.555 in wet conditions.

He was joined on the front row of the grid by Kovalainen. Mark Webber qualified third. Massa took sixth position behind

Metanephrops challengeri is a species of slim, pink lobster that lives around the coast of New Zealand. It is 13–18 cm long and weighs around 100 g; the carapace and abdomen are smooth, adults are white with pink and brown markings and a conspicuous pair of long, slim claws. M. challengeri lives in burrows at depths of 140–640 m in a variety of sediments. Although individuals can live for up to 15 years, the species shows low fecundity, where small numbers of larvae hatch at an advanced stage. M. challengeri is a significant prey item for ling, as well as being an important fishery species for human consumption. The species was first collected by the Challenger expedition of 1872–1876, but only described as separate from related species by Heinrich Balss in 1914. Although classified in the genus Nephrops, it was moved in 1972 to a new genus, along with most other species classified in Nephrops. Metanephrops challengeri is a slender lobster 13–18 centimetres long, but exceptionally up to 25 cm, weighing up to 100 grams each.

Its chelipeds are long and unequal. The second and third pairs of pereiopods end in small claws, but the fourth and fifth pairs do not; the carapace is smooth, extends forwards into a long, narrow rostrum, only shorter than the carapace. Adults are white, but the front half of the rostrum, the sides of the abdomen, are pink. Bright red bands extend across the base of the rostrum, the posterior edge of the carapace, the chelipeds, each of the abdominal segments; the dorsal parts of the abdomen are brown, there are two brown saddles on the dorsal carapace. M. Challengeri is considered to have the most primitive morphology of any species of Metanephrops, having fewer novelties than the oldest known fossil species, M. rossensis. Its rostrum is longer than that of other species in the thomsoni species group, the ridge along the midline of the carapace only has two small spines. Unlike some other species of Metanephrops, the carapace is smooth, as are the abdominal tergae, the chelipeds are covered in fine granules.

Metanephrops challengeri reaches sexual maturity at the age of 3–4 years, may live up to 15 years in total. Females produce large eggs in small numbers; the larvae hatch at the zoea stage. The zoea larvae are 10.0–11.5 mm long, possess all the appendages of the cephalothorax, including the pereiopods, which are used for swimming, but no pleopods. This larval stage lasts less than four days, before the young moult into the post-larval stage; the post-larva swims using its pleopods. The post-larva moults into the adult form. Larvae are seen in the wild, confirming that the development to the bottom-dwelling post-larva is rapid. Metanephrops challengeri lives around the coasts of New Zealand, including the Chatham Islands, at depths of 140–640 metres, it lives in burrows in a variety of "suitable cohesive" sediments, is a significant prey item for ling. Lobsters have few parasites, the most important for M. challengeri being the microsporidian Myospora metanephrops. This can cause "destruction of the skeletal and heart muscles of infected lobsters", but its significance for the animals and for the fishing industry remains unclear.

When it was described in 2010, M. metanephrops was the first microsporidian to be isolated from a true lobster. Metanephrops challengeri has been harvested commercially since the 1980s. Between the season of 1988/89 and 1990/91, the amount of scampi caught around New Zealand increased from only 55,000 kilograms to around 500,000 kg. Catch limits were introduced in 1990/91, now 1,000,000 kg is caught annually by trawlers; the fishery is centred on four areas of continental shelf of the submerged continent Zealandia: the Campbell Plateau around the Auckland Islands, Chatham Rise, along the Wairarapa coast, in the Bay of Plenty. Most of the fishing vessels used to capture M. challengeri are 20–40 metres long, with "double or triple trawl rigs of low headline height". There is considerable variation in the catch per unit effort between different depths, between different geographical areas and between different years. M. challengeri is considered a luxury foodstuff. Most of the catch is exported and as a result, it is seen in restaurants in New Zealand.

Metanephrops challengeri was the subject of a 2003 select committee inquiry in the New Zealand parliament, after allegations of corruption arose against officers of the Ministry of Fisheries. Although the allegations were quashed, the inquiry ruled that preferential treatment had been given to the large fishing company Simunovich Fisheries. In response, the government introduced M. challengeri into their Quota Management System and paid compensation to some fishermen who had a justified grievance. Under QMS, an overall limit of 1,291,000 kg was put in place for M. challengeri in 2011. Metanephrops challengeri is listed as Least Concern on the IUCN Red List, due in part to the Quota Management System put in place by the New Zealand government; the species does appear to be declining, based both on burrow counts and analyses of catch per unit effort. Estimates of the total population size of M. challengeri vary dependin