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List of logarithmic identities

In mathematics, there are many logarithmic identities. Logarithms and exponentials with the same base cancel each other; this is true because exponentials are inverse operations. B log b ⁡ = x because antilog b = x log b ⁡ = x because log b ⁡ = x Both of the above are derived from the following two equations that define a logarithm: b c = x, log b ⁡ = c Substituting c in the left equation gives blogb = x, substituting x in the right gives logb = c. Replace c with x. Logarithms can be used to make calculations easier. For example, two numbers can be multiplied just by adding; the first three operations below assume x = bc, and/or y = bd so that logb = c and logb = d. Derivations use the log definitions x = blogb and x = logb. Where b, x, y are positive real numbers and b ≠ 1. Both c and d are real numbers; the laws result from canceling appropriate law of indices. Starting with the first law: x y = b log b ⁡ b log b ⁡ = b log b ⁡ + log b ⁡ ⇒ log b ⁡ = log b ⁡ = log b ⁡ + log b ⁡ The law for powers exploits another of the laws of indices: x y = y = b y log b ⁡ ⇒ log b ⁡ = y log b ⁡ The law relating to quotients follows: log b ⁡ = log b ⁡ = log b ⁡ + log b ⁡ = log b ⁡ − log b ⁡ log b ⁡ = log b ⁡ = − log b ⁡ Similarly, the root law is derived by rewriting the root as a reciprocal power: log b ⁡ = log b ⁡ = 1 y log b ⁡ log b ⁡ a = log 10 ⁡ log 10 ⁡ This identity is useful to evaluate logarithms on calculators.

For instance, mo

Convention on the Law Applicable to Products Liability

The Convention on the Law Applicable to Products Liability is a convention concluded in 1971 within the framework of the Hague Conference on Private International Law, which governs the law that should be applied to products liability cases. It entered into force in 1973 and as of 2020, 11 countries are party to it; the convention determines the law that applies between the manufacturer or distributor of a product and the person that have suffered damage. It thus does not handle which court has jurisdiction, or whether a judicial decision should be recognized; the convention does not apply between the seller that directly sold the product to the person suffering the damage. The exclusion was chosen as the negotiators deemed the relationship between those two parties clear enough, would hamper wide adoption of the convention; the products the convention applies to "shall include natural and industrial products, whether raw or manufactured and whether movable or immovable", thus has a wide meaning.

Member states can decide to exclude agricultural products from the scope of the convention, as Spain did. The convention's main articles determining which law should be applied are Articles 4 and Article 5, they are summarized below

Salticus cingulatus

Salticus cingulatus is a Palearctic jumping spider of the family Salticidae. The female Salticus cingulatus has a body length of 5.2 -- 5.5 mm. On the male the palpal bulb is distally forked, with a long visible part. On the female the epigyne is sclerotized, the anterior part opaque; the prosoma is black, with white spots, the opisthosoma is black, but covered with white hairs. It is similar to Salticus scenicus but it is distinguished by being much paler due to more white hairs on the prosoma; the favoured habitat of Salticus cingulatus is old tree trunks and fence palings situated in sunny situations in or close to woodland and heathland. It is encountered on pines, it is encountered on buildings. Males are active in May and June, females between May and July, but they persist until autumn. Salticus cingulatus has a Palearctic distribution, it is found throughout Europe except Iceland. It is a distributed species in Great Britain but with only scattered records on the west, south-east and the far north

Bari Centrale railway station

Bari Centrale is the main railway station of the Italian city of Bari, capital of Apulia. It is one of the most important railway stations in Italy, with an annual ridership of 14 million; the station was first built in 1864 and between 1865 and 1906 5 more platforms were added. In the first half of the 20th century it was further enlarged and renewed and in 1946, after World War II, the most recent renovation project was started. Bari Centrale lies at Aldo Moro square, it is a large junction station with 16 platforms for passenger service. The station is situated on the lines Ancona-Pescara-Bari-Lecce, Bari-Taranto, Bari-Martina Franca-Taranto; the station is divided into 3 areas. The main station features 16 platforms. On the west side of Piazza Aldo Moro are the entrances to the Ferrotramviaria and Ferrovie Appulo Lucane stations, with 3 and 2 platforms respectively; the Centrale is an important transportation hub for the Apulia regional services. For long-distance transport it is served by EuroStar, InterCity and Express trains to Rome, Bologna and Venice.

Until 1994 it was served by the express "Parthenon" Athens-Paris. The station is served by the following services: High speed services Milan - Bologna - Ancona - Pescara - Foggia - Bari High speed services Rome - Foggia - Bari - Brindisi - Lecce High speed services Milan - Parma - Bologna - Ancona - Pescara - Foggia - Bari - Brindisi - Lecce High speed services Milan - Parma - Bologna - Ancona - Pescara - Foggia - Bari - Taranto High speed services Turin - Parma - Bologna - Ancona - Pescara - Foggia - Bari - Brindisi - Lecce High speed services Venice - Padua - Bologna - Ancona - Pescara - Foggia - Bari - Brindisi - Lecce Intercity services Rome - Foggia - Bari Intercity services Bologna - Rimini - Ancona - Pescara - Foggia - Bari - Brindisi - Lecce Intercity services Bologna - Rimini - Ancona - Pescara - Foggia - Bari - Taranto Night train Rome - Foggia - Bari - Brindisi - Lecce Night train Milan - Parma - Bolgona - Ancona - Pescara - Foggia - Bari - Brindisi - Lecce Night train Milan - Ancona - Pescara - Foggia - Bari - Taranto - Brindisi - Lecce Night train Turin - Alessandria - Bolgona - Ancona - Pescara - Foggia - Bari - Brindisi - Lecce Regional services Bari - Brindisi - Lecce Regional services Foggia - Barletta - Bari Local services Bari - Gioia del Colle - Taranto Local services Bari - Conversano - Putignano - Martina Franca Local services Bari - Casamassima - Putignano Local services Bari - Altamura - Gravina - Potenza Local services Bari - Altamura - Matera Bari Metropolitan services Bitonto - Palese - Bari Bari Metropolitan services Barletta - Andria - Bitonto - Aeroporto - Bari Bari Metropolitan services Ospedale - Bari Bari Metropolitan services Bitonto - Aeroporto - Bari 100 Bari - Valenzano - Adelfia - Putignano - Alberobello - Martina Franca 110 Bari - Casamassima - Gioia del Colle - Taranto 120 Bari - Triggiano - Capurso - Cellamare - Casamassima - Sammichele 130 Bari - Conversano - Putignano - Noci - Taranto 150 Bari - Monopoli - Fasano - Brindisi Bari metropolitan railway service List of Bari metropolitan railway stations Railway stations in Italy

PNC Second Street Market

The 2nd Street Market is a public market in Dayton, Ohio. The market is located at the corner of East 2nd Street, it is Dayton's oldest operating public market. In 2012, Country Living Magazine mentioned the market in its piece called “50 Things To Do This Summer in 50 States”; the market is operated by Five Rivers MetroParks. The market began as an extension of the Wegerzyn Outdoor Farmers Market, which moved to the Cannery building on Wayne Avenue to provide an indoor setting and to extend their season in the late 1990s. In November 2001, the market moved to its present location in a former Baltimore and Ohio Railroad warehouse built in 1911; the building was saved from demolition and renovated in 2001 in partnership with Five Rivers Metro Parks, Webster Station Development Group, City of Dayton and PNC Bank. The 2nd Street Market attracts 370,000 people each year; the market is open three days a week year-round Thursday and Saturday. Because the market is, for the most part, the weather does not affect the hours of operation though crowds are biggest during the summer months.

Additional outside stalls are set up in the warm weather months. Local produce and baked goods are the most popular and available items at The 2nd Street Market. In-season produce is always for sale, including fruits and eggs. A variety of cheeses are offered at several stands as well as foods such as honey, maple syrup, coffee beans. Fresh, homemade baked goods dominate the morning hours and are enjoyed by visitors as a breakfast treat. Organic items and meats are available at some stalls; some stands offer luncheon fare including full platters. The market contains more than 50 different vendors. Entertainment at The 2nd Street Market occurs year-round and marks special occasions such as Mardi Gras, St. Patrick's Day, or Christmas. Note: Images were taken on a non-operating day. 2nd Street Market Website

Astrakhan Kremlin

Astrakhan Kremlin is a fortress in Astrakhan, Russia. It is located on a hill on an island in the Volga Delta, between the Volga, the Kutum, the Tsarev. For centuries, the Astrakhan Kremlin was inapproachable stronghold in the south-eastern border of the Russia. Series of historical events are related to the fortress: the Crimean Turkish hiking on the lower Volga in the 16th century, the " Troubles " in Russia and the peasant uprising led by Stepan Razin in the 17th century, the transformation of the tsar Peter the Great Period, revolt of archers in 1705-1706, development of the Caspian navy in the 18th century, the reinforcement of the country's borders and entry into a part of Russian territories of the Caucasus and Central Asia. In 1552, Tsar Ivan IV conquered the Kazan Khanate. Four years Russian troops took Astrakhan; when the Middle and Lower Volga Region was joined, Moscow department acquired access to the Caspian Sea, in the new lands began to emerge and strengthen the Russian city. The first construction of the Kremlin began in 1587- 1588 under the guidance of sapper, a lector of Discharge Order – I.

G. Vorodkov, he towers. The place of construction was chosen on the hill, known as “Rabbit” or “Zayachii” in Russian, it was the best place to build the fortress, because from the west and the north the hill was surrounded by the Volga River, south and east – by lakes and marshes. During the reign of Ivan IV The Terrible and Boris Godunov the wooden fortress was rebuilt into a stone one. For the development of Kremlin walls and towers state-owned official masters were headed from Moscow to Astrakhan; the chiefs of creation and planning of stone fortress were Mikhail Ivanovich Veliyaminov, Grigorii Ovcin and lecturer Dei Gubastii. For best results executives used the old, but strong Tatar plinths which were brought from the ruins of the cities of the Golden Horde towns. Stone citadel was built by the type of Moscow Kremlin; the walls of the Astrakhan Kremlin were equipped with prongs the slots of which allowed to fire from handguns against enemies. At the same place at the top tier were formed "Varnitsa"—holes through which boiled water or hot tar was poured on attackers.

During those times the system of organization of “fire fighting” of the fortress was invoked as one of the most modern systems in whole Russia. Construction of Assumption Cathedral, which began in 1699 on the first of October, was the most significant event, held on the territory of the Astrakhan Kremlin. Thirty professional stonemasons were invited led by bondman architect Dorotheos Myakisheva. Building of the Cathedral lasted 12 years. During this time, the architects created the temple with proper cubic form with five heads; the external of the Cathedral was carved with white stone. Windows and dome heads were framed by columns in the style of Corinthian décor and semicircular arches were filled with paintings with biblical plot. Three of such arches were arranged on each side of the temple; the cathedral was divided into two floors: the upper church is dedicated to the honor of the Assumption of the Blessed Virgin. Tall and light temple was intended for ceremonial worships during warm months.

The lower church, dark lightened and surrounded by the gallery columns. In 1710 the bell tower cathedral was constructed under the control of architect Dorothea Myakisheva. In addition to bells, the watch was installed, which created freshness and uniqueness in the city; the next two centuries were calm for the Kremlin. Its buildings were repaired and renewed. At the beginning of the 20th century, after the October Revolution, access to the Territory of Kremlin was closed. Instead it was transformed as a military post, where groups of Red Guards were formed the Military Revolutionary Committee was placed. In January 1918 Astrakhan Kremlin was once again in the middle of fateful events, when supporters of Soviet power fought with Astrakhan Cossacks, they attacked The Red Army, entrenched in the Kremlin, from the roofs of nearby buildings. Serious destruction was caused to the Kremlin after this battle. Miraculously the temples of Kremlin have survived. In 1919 the Army was reorganized under the leadership of Kirov to protect the outfall of Volga and to defeat the White Guard troops and foreign interventionists.

Thus, in the early 20th century the Kremlin remained a military target. It was popularly named as “The Town of Trotsky”. Only after the end of World War II did the town open access to the Kremlin. At the same time, the Kremlin ceased to be subject of military purposes. In the middle of the 20th century, significant restoration works were held, due to which many buildings, requiring urgent repairs were saved. In 1974 the Astrakhan Kremlin became a museum and in 1980 became part of the Astrakhan State United Historical-Architectural Museum-Reserve. Nowadays citizens and tourists of Astrakhan have access to museum exhibits of the lifestyle of the Astrakhan Garrison. Moreover, they can see Casual Suits archers and scorers, elements of their weapons and ammunition, the exhibition dedicated to the history of popular uprisings and corporal punishment. In 2011, after the restoration of the Kremlin, the Guardhouse exposition was opened, which tells visitors about the life of Astrakhan military garrison of the 19th century.

Walls of Kremlin, 1582—1589 The cathedral bell tower with Prechistinsky Gate Archbishop Tower Zhitnyaya Tower, 1582—1589 Crimean Tower'Red Gate' Tower Water Gate St. Nicholas Church Gate, 1729—1738 Artillery Tower and Yard (now it is museum with the s