Logopolis is the seventh and final serial of the 18th season in the British science fiction television series Doctor Who, which was first broadcast in four weekly parts from 28 February to 21 March 1981. It was Tom Baker's last story as the Doctor and marks the first appearance of Peter Davison in the role. This serial also marks the first appearance of Janet Fielding as new companion Tegan Jovanka, whilst Nyssa, played by Sarah Sutton and seen in previous serial The Keeper of Traken, returns and also joins the Doctor as a companion.

Warned of upcoming danger by the TARDIS' Cloister Bell, the Doctor decides to stay out of trouble, and instead spends time fixing his broken chameleon circuit by extracting the measurements of a true police box on Earth in order to give to the mathematicians of the planet Logopolis. He is unaware that the Master is aware of his plan, and materialises his own TARDIS in a recursive loop with the Doctor's. The Doctor and Adric work out the recursion, and attempt to flush the Master out by materialising the TARDIS in the River Thames, but instead land on a boat on the river with no sign of the Master's TARDIS. Outside, the Doctor sees a glowing white figure, the Watcher, who directs him to hurry to Logopolis. In flight, they discover they have picked up a passenger, Tegan Jovanka, an airline stewardess who wandered in the TARDIS to call for help to fix her aunt's flat tyre; the Doctor informs her the Master has killed her aunt with his Tissue Compression Eliminator.

Thomas Stewart "Tom" Baker (born 20 January 1934) is a British actor. He is best known for playing the fourth incarnation of the Doctor in the science fiction television series Doctor Who, a role he played from 1974 to 1981 (he is to date the longest serving actor to play the part). In later years Baker played the roles of Professor Geoffrey Hoyt in Medics (1992-95) and Donald Ulysses MacDonald in Monarch of the Glen (2004-05). He also narrated the sketch show Little Britain from 2003 to 2006.

Baker was born in Scotland Road, Liverpool, Lancashire, the son of Mary Jane (née Fleming), a cleaner and John Stewart Baker, a sailor who was rarely at home. His parents were working class Liverpudlians. Baker left school at 15 to become a Roman Catholic monk and remained in the monastic life for six years, but left after losing his faith, and did his national service in the Royal Army Medical Corps, serving for two years from 1955 until 1957. At the same time he took up acting, at first as a hobby and finally turning professional towards the end of the 1960s.

The Fourth Doctor is the fourth incarnation of the protagonist of the long-running BBC British television science-fiction series Doctor Who.

He was portrayed by Tom Baker for seven consecutive seasons from December 28, 1974 (Part One of Robot) to March 21, 1981 (Part 4 of Logopolis), and remains the longest-lived incarnation of the Doctor in the show's on-screen history, counting both the classic and modern series.

Within the series' narrative, the Doctor is a centuries-old Time Lord alien from the planet Gallifrey who travels in time and space in his TARDIS, frequently with companions. When the Doctor is critically injured, he can regenerate his body, changing his physical appearance and personality in the process.

The Fourth Doctor appeared in 172 episodes (179, counting the regeneration in Planet of the Spiders and the aborted Shada) over a seven-year period, from 1974 to 1981. This makes him the longest running on-screen Doctor of both series. He also appeared in the specials The Five Doctors (via footage from the incomplete Shada) and made his final appearance as the Doctor in the charity special Dimensions in Time (aside from a series of television advertisements in New Zealand in 1997).

Peter Davison (born Peter M. G. Moffett on 13 April 1951) is a British actor, best known for his roles as Tristan Farnon in the television version of James Herriot's All Creatures Great and Small and the fifth incarnation of the Doctor in Doctor Who, which he played from 1982 to 1984.

Davison was born Peter Moffett in Streatham, London, son of an electrical engineer who was originally from Guyana. The family then moved to Knaphill in Surrey. Before becoming an actor, he gained three O-levels at Winston Churchill School, St John's, Woking, Surrey, and then had several odd jobs, including a stint as a mortuary attendant.

Davison studied at the Central School of Speech and Drama. His first job was as an actor and assistant stage manager at the Nottingham Playhouse. He chose the stage name Peter Davison to avoid confusion with the actor and director Peter Moffatt, with whom Davison later worked.

His first television work was in a 1975 episode of the children's science fiction television programme The Tomorrow People, alongside American actress Sandra Dickinson, whom he married on 26 December 1978. Davison portrayed an alien named "Elmer", often tortured with "tickling boots" under the control of his sister (played by Dickinson) and his mother (played by Margaret Burton) known as "the Mama".

Paddy Kingsland (born 30 January 1947) is a composer of electronic music best known for his incidental music for science fiction series on BBC radio and television whilst working at the BBC Radiophonic Workshop. Educated at Eggars Grammar School, Alton, in Hampshire, he joined the BBC as a tape editor before moving on to become a studio manager for BBC Radio 1. In 1970 he joined the Radiophonic Workshop where he remained until 1981. His initial work was mostly signature tunes for BBC radio and TV programmes before going on to record incidental music for programmes including The Changes, two versions of The Hitchhiker's Guide to the Galaxy: the second radio series and the TV adaptation, as well as several serials of Doctor Who. His work on the latter series included incidental music for the following serials (in order of transmission):

Other well-known series which contained music composed by Paddy Kingsland are Around the World in 80 Days and Pole to Pole, both travel series by Michael Palin. Paddy Kingsland also composed music for many schools' television series including Words and Pictures, Rat-a-tat-tat, Watch, Numbercrew, Storytime, English Express, Music Makers, Hotch Potch House and the Look and Read stories "Joe and the Sheep Rustlers" and "The Boy from Space". Since leaving the BBC, composed music for the KPM music library, television, commercials and corporate videos. He also owns his own studio, PK Studios. He is currently composing the music for the CITV series Blips produced by Ragdoll Productions.