The 32-year-old Farah has been under the spotlight since the American was accused of violating anti-doping rules in a BBC documentary earlier this month.
The BBC programme Panorama, in association with American website ProPublica, made a series of allegations including that Salazar had given Olympic 10,000 metres silver medallist Galen Rupp the banned anabolic steroid testosterone.
Salazar, who issued a lengthy and detailed denial of the allegations on Wednesday, has worked with Farah since 2011 and operated as a consultant for British Athletics since 2013.
"Following all the speculation I want to make it clear where I stand," said Farah who was not accused of any wrongdoing in the BBC documentary.
"Although it's been a difficult time I asked Alberto to respond to the allegations made against him and he has now done so in full," added the Briton in a statement on his Facebook page.
"As someone I've worked with for many years I feel I have to believe in Alberto and the evidence he has provided.
"Based on that evidence I will continue to work with him and hope now that I can focus on what I do best, training hard to win medals for my country. Thanks."
UK Athletics CEO Niels de Vos added: “British Athletics note and respect Mo Farah’s decision to continue his coaching relationship with Alberto Salazar.
"As has always been the case, British Athletics will continue to take responsibility for all nutritional, medical and sport science elements of Mo Farah’s training regime.”
Farah is due to return to competition at the Monaco Diamond League meeting on July 17.
(Reporting by Toby Davis, editing by Tony Jimenez)
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