Clear[u, lag, eqx, eqy, eqz, eql, result1, v]

u[x_, y_, z_] :=  a Log[x]  +  b Log[y]  +  c Log[z];

lag[x_, y_, z_, l_] :=  
      u[x, y, z]  +  l (m  -  p x  -  q y  -  r z);

\(eqx = D[lag[x, y, z, l], x] == 0; eqy = D[lag[x, y, z, l], y] == 0; 

    eqz = D[lag[x, y, z, l], z] == 0; eql = D[lag[x, y, z, l], l] == 0; \)

result1  = Solve[{eqx, eqy, eqz, eql}, {x, y, z, l}]

v[p_, q_, r_, m_] := u[x, y, z] /. result1;

Map[D[v[p, q, r, m], #1]&, {p, q, r, m}] // Simplify

{- l x,  -l y,  -l z} /. result1