2 of 2 solution using FindIntegerNullVector

I am interpreting your question to mean that you want logarithms of rational numbers to be expressed purely in terms of logarithms of primes. If so, one can use a replacement rule:

{Integrate[1/(x Sqrt[x + 4]), {x, 5, 21}], 
 Integrate[1/(x Sqrt[x + 16]), {x, 9, 33}]} // Simplify
   {1/2 Log[15/7], 1/4 Log[27/11]}

% /. Log[r_Rational] :> (Total[#2 Log[#1] & @@@ FactorInteger[Numerator[r]]] - 
                         Total[#2 Log[#1] & @@@ FactorInteger[Denominator[r]]]) // Expand
   {Log[3]/2 + Log[5]/2 - Log[7]/2, 3 Log[3]/4 - Log[11]/4}

Alternatively, one can use FindIntegerNullVector[], similar to what was done in this answer:

-Rest[#]/First[#] &[FindIntegerNullVector[{1/2 Log[15/7], Log[3], Log[5], Log[7]}]]
   {1/2, 1/2, -1/2}

-Rest[#]/First[#] &[FindIntegerNullVector[{1/4 Log[27/11], Log[3], Log[11]}]]
   {3/4, -1/4}