I have the following problem:
Find conditions on the real numbers $a<0,b>0,c>0,d>0$ such that the function $$x→a×\log(b+cx)+d$$ has a fixed point $x<0$. A similar question can be found in Conditions on $a>0,b>0,c>0,d<0$ such tha $x→a×log(b+cx)+d$ has a fixed point $x>0$
However, the method is not applicable for this case.
Since $x$ is increasing and $a\ln(b+cx)+d$ is decreasing (because $\ln$ is increasing and $a<0$), the difference $f(x)=x-[a\ln(b+cx)+d]$ is increasing. Its domain is $(-b/c,\infty)$ which contains $0$, so it suffices to simply check the sign of $f(0)=-a\ln b-d$. If $f(0)\ge0$ then $f(x)>0$ for all $x>0$, but if $f(0)<0$ then there must exist an $x>0$ for which $f(x)=0$ (since $\lim\limits_{x\to\infty}f(x)=\infty$).
The same logic can be used to determine if there is a fixed point in $(-b/c,0)$.
Indeed, the difference $f(x)=x-[a\ln(b+cx)+d]$ is increasing and continuous. If $f(0)=-a\ln b-d>0$ then there must exist an $x<0$ for which $f(x)=0$ (since $\lim\limits_{x\to -b/c}f(x)=-\infty$).