<s>[INST] What can $\log_e(x) + \log_e(1+x) =0$ be written as? The equation $\log_e(x) + \log_e(1+x) =0$ can be written as:
a) $x^2+x-e=0$
b) $x^2+x-1=0$
c) $x^2+x+1=0$
d) $x^2+xe-e=0$
I tried differentiating both sides, then it becomes $\frac{1}{x}+\frac{1}{1+x}=0$, but I dont get any of the answers.
 [/INST] Since $\log(A) + \log(B) = \log(AB)$
$$
\begin{align}
\log(x) + \log(1+x) &=0\\
\Rightarrow \log{\left( x(1+x)\right)} &=0 \\
\Rightarrow x(1+x) &= 1 \\
\Rightarrow x^2 + x -1 &=0
\end{align}
$$
</s>