Can an entire $f$ satisfy $x>k | f(x+yi)=\ln(x+yi+z)+o(1) $?

Question

Let $z$ be a complex number. Let $i$ be the imaginary unit. Let $x,y,k$ be positive real numbers.

Consider

$$x>k | f(x+yi)=\ln(x+yi+z)+o(1) $$

true for all $x>k,y$ and some $k,z$.

Is there an entire function $f$ that satisfies this ?