On the right half-plane, what is an upper bound for $\frac{1}{\log(z+2)}$?

Question

I am trying to estimate some factors in my integrand in complex integration, and I think the upper bound for $\frac {1}{log(z+2)}$ on the semicircle in the right half plane is just $\frac {1}{log(2)}$. Anything wrong with this upper bound? I'm probably just overthinking this one, but wanted to make sure.

My parameterization is:

z= iy, $-R \le y \le R$ on the imaginary axis

z= $Re^{i\theta}$ on the semi-circle in the RHP,

and I let R grow to infinity.

Thanks,