Mujahid, speaking by phone from an undisclosed location, claimed that 20 Afghan police were killed in the attacks.
Logar, located some 60 km south of Kabul, has witnessed increased Taliban activities in the past two years. Some 1,500 US soldiers were deployed to the province earlier this year to prevent the militants' advancement towards Kabul.
Taliban militants have recently launched several attacks in southern and eastern regions involving including suicide bombers, who simultaneously attacked multiple government buildings.
In the recent series of such attacks, eight Taliban fighters, equipped with guns and suicide vests, attacked Gardez and Nangarhar, two capital cities in the eastern region on July 21, killing five Afghan security personnel.
Meanwhile, nine Taliban militants were killed in a US-led coalition airstrike in Charchino district of southern Uruzgan province Monday, Musa Khan, spokesman for the provincial police chief said.
He said the airstrike was conducted after the militants attacked a convoy of the forces in the area. There were no casualties among the foreign forces, but three Afghan security forces, who were accompanying the coalition convoy, were wounded.
Thousands of Afghan and international forces are taking part in various operation in southern region to flash out the militants from the districts they are holding in order to provide a safer environment for the August 20 presidential election.
There are more than 100,000 international forces and around 200,000 Afghan security forces in the country to provide security for polling day, the second direct chance for Afghans to elect their president in country's recent history.