Isaf did not specify the nationality of the captured soldiers, but US officials told the Associated Press that the soldiers are Americans.
A Taliban spokesman who earlier announced the capture also described the soldiers as American. He told the Reuters news agency that three US soldiers were initially captured, but that one of them has died.
'They stopped in the main bazaar'
The governor of the province told Al Jazeera that he believes the missing Americans work for a Provincial Reconstruction Team, a military-led development team.
Samer Gul, the administrative head of Charkh district in Logar, said the Taliban noticed the soldiers when their vehicle passed through a market.
"They stopped in the main bazaar of Charkh district. The Taliban saw them in the bazaar," Gul said. "They didn't touch them in the bazaar, but notified other Taliban that a four-wheel vehicle was coming their way."
Gul said the soldiers were captured following a shootout with the Taliban.
A journalist from Reuters reported hearing local radio stations in Logar broadcasting US statements that offered a $20,000 reward for information about the captured Americans.
Logar is southeast of Kabul, the Afghan capital, in an area that has seen growing insurgent activity over the last few years.
Captures of foreign soldiers are rare in Afghanistan. The only other soldier known to have been captured is Bowe Bergdahl, an American who was captured by the Taliban in Paktika province in June 2009. Paktika is near Logar in eastern Afghanistan.
His fate and whereabouts remain unclear, though he appeared in a video released by the Taliban in April.