Difference equation with log

Question

Can we find continuous $f(x)$ and $d$ such that $$ f(x+1)-f(x) = -\log( c|x| + d ) $$ for all $x$? The constant $c>0$ is specified.

Answer 1

$f(x+1)-f(x)=-\log(c|x|+d)$

$f(x)=\Theta(x)-\sum\limits_x\log(c|x|+d)$ , where $\Theta(x)$ is an arbitrary periodic function with unit period

$f(x)=\Theta(x)-\log\prod\limits_x(c|x|+d)$ , where $\Theta(x)$ is an arbitrary periodic function with unit period