The son of Levy Salomons of St Mary Axe and Frant, Sussex and Matilda de Metz of Leyden (married in 1795), he followed his father into business in the City of London, where he was a successful banker. Salomons was one of the founders of the London and Westminster Bank (now the NatWest), and a member of the London Stock Exchange.
Salomons was educated for a commercial life and in 1832 was one of the founders of the London and Westminster Bank. He began as an underwriter on his own account in 1834, being one of the few Jews to participate in the development of joint-stock banking in Britain.
In 1835 David Salomons was chosen as one of the sheriffs of London and was able to accept the position following a special act of Parliament making it possible to admit Jews to certain municipal offices by administering “such an oath as would be binding on their conscience.” When he ended his term of office a silver testimonial was presented to him by the Jewish community “as an acknowledgment of his exertions in the cause of religious liberty.”
However, when he was elected alderman for the ward of Aldgate, he refused to take the Christian oath, and was forced to withdraw. He continued to fight for the abolition of Jewish disabilities, but when reelected in 1844. was still unable to take office. It was only after an act of Parliament was passed the following year enabling Jews to hold municipal offices that Salomons’ election as alderman in 1847 was finally recognized.
Salomons had in the meantime held other offices, being appointed high sheriff of Kent in 1839-1840, deputy lieutenant for Kent, Sussex, and Middlesex, and in 1838 the first Jewish magistrate for Kent.
In 1851 Salomons was elected to Parliament as the Liberal representative of Greenwich, but declined to take the oath “on the true faith of a Christian." However, he voted three times without having been sworn in the statutory manner. Prolonged legal proceedings followed and he w as fined five hundred pounds and debarred from the house. After the parliamentary oath was amended in 1858, he was again elected for Greenwich and continued to represent that constituency until his death.
In 1855 Salomons was elected lord mayor of London and upon conclusion of his term of office he received the unique distinction of an address of congratulations signed by the leading merchants and bankers of the city.
Salomons published a number of articles on financial and monetary matters, as well as an account of the persecution of the Jews in Damascus in 1840. He was active in the Board of Deputies ot British Jews and took an interest in the Westminster Jews' Free School, the Jews' Hospital, and the Society for Hebrew Literature. He was fond of art and had an excellent collection of modern paintings.
In 1S69 he was created baronet, being succeeded by his nephew David Lionel Salomons (1851-1925), electrical engineer and pioneer of horseless carriages. He attended University College of London, and took a degree in natural sciences at Cambridge.
In 1874 he claimed to have been the first person to use electric lighting with incandescent lamps. He also made and used home-made electric carriages, took out patents on many electrical inventions, and published a number of books on the subject. He served as treasurer and vice-president of the Council of the Institution of Electrial Engineering and was one of the foundersofthc Royal Automobile Club and the Aero Club de France.
Salomons married in 1825 Jeanette, daughter of Solomon Cohen of Canonbury House and Hannah Samuel. Her aunts Judith and Henriette were the wives of Sir Moses Montefiore and Nathan Mayer Rothschild respectively. After her death in 1867 Salomons married Cecilia, the daughter of Samuel Moses Samuel in 1872. He was made a baronet of Broom Hill in the parish of Tonbridge in the County of Kent and of Great Cumberland Place in the County of Middlesex on 26 October 1869.