Solving $ f'(x) =-\log( f(x) +a ) $

Question

Can the solution of $$ f'(x) = -\log( f(x) + a ) $$ with $f(0)=0$ and $a \in (0,1)$ be well approximated by the Lambert W function for $x>0$? It seems that morally this might be the case (by relating $f(x)$ to $f'(x)$ and bootstrapping).

Answer 1

$$ \frac{d}{dx}f(x)=-\ln (f(x)+a) $$ $$ \frac{d}{dx}\left[f(x)+a\right]=-\ln (f(x)+a) $$ $$ \frac{\frac{d}{dx}\left[f(x)+a\right]}{\ln(f(x)+a)}=-1 $$ $$ \frac{\frac{d}{dx}\left[f(x)+a\right]}{\ln(f(x)+a)}dx=-1\ dx $$ $$ \int\frac{1}{\ln(f(x)+a)} d\left[f(x)+a\right] =-\int dx $$ $$ \mathrm{li}(f(x)+a)+C_1=-x+C_2 $$ $$ \mathrm{li}(f(x)+a)=-x+C $$ Where $\mathrm{li}(x)$ is the logarithmic integral function.