Company Names, addresses, dates:
  Ballston Pharmacy, Stuart St. & Fairfax Dr, Ballston VA (1906-ca.1915) ^1,2
  Ballston Drug Store, William L. White prop., Stuart Street, Arlington VA (ca.1915-1924) ^2,3
  Ballston Drug Store, Stuart Street, Arlington VA (ca.1924-ca.1936) ^3,4
  Ballston Drug Store, R.C. Osterwise Prop., 4227 Fairfax Drive, Arlington VA (ca.1936-ca.1961) ^4,5

Notes:
Dr. Williamson Crothers Welburn (1874-1964)⁶ came to Ballston in 1905 and practiced medicine. He established the Ballston Pharmacy, which was the first drug store in Arlington County, established before it was even called Arlington County. Prior to that, residents had to go all the way to Georgetown to fill prescriptions.⁷

Circa 1915, Mr. William L. White (1874-1925)⁸ became the owner of the drugstore.²
In 1924, he sold the store to John W. Rogers, who was the president of the O'Donnel Drug Company in Washington DC.³
During the years that William White was owner, newspaper mentions refer to the store as both "Ballston Pharmacy" and "Ballston Drug Store", so I assume this was the same store that was stared by Dr. Welburn.
At that time, the drugstore was located on "Ballston Avenue", which according to Wikipedia was renamed "Stuart Street".⁹

Circa 1936, Richard C. Osterwise (1887-1961)⁵ became the proprietor of the Ballston Drug Store⁴.
Newspaper articles and ads that mention the location of the drug store in these later years list it at 4227 Fairfax Drive, which is roughly on the corner of Fairfax Drive and Stuart Street, so it's possible that the location of the store was constant since it was started by Dr. Welburn.
Osterwise passed away in 1961, and I couldn't find any newspaper mentions of the Ballston Drug store after that, so I assume it closed down about that time.

This is a 7"x3.5" calendar from 1948, that says: (picture of boy with fishing rod & dog) / ... Ballston Presc. Drug Store / 4227 Fairfax Drive / Arlington, Virginia / TUMS FOR THE TUMMY ...