We determine a necessary and sufficient condition for the infinitude of primes p such that none of the equations ​​​​​​​a^x_i ≡ b_i (mod p), 1 ≤ i ≤ n, are solvable. We control the insolvability a^x ≡ b(mod p)of by power residues for multiplicatively independent a and b, and by divisibilities and, most importantly, parities of orders in multiplicatively dependent cases. We also consider a more general problem concerning divisibilities of orders. The problems are motivated by Artin's primitive root conjecture and its variants.