Proof for the upper bound on entropy $H(S)$?

Question

I was trying to prove the upper bound on $H(S)$ using the inequalities $\ln(x)\le(x-1)$ and $\ln(1/x)\ge(1-x)$ for independent and memory less source symbols $s_1,\dots,s_q$ . I am trying to prove $\log(q)\ge H(S)$