Solve xlog_{e}({x+sqrt{1+x^2}})=xlog_{e}({-x+sqrt{a+x^2}})

$$x \log_{ e }({ x+ \sqrt{ 1+ { x }^{ 2 } } }) = x \log_{ e }({ -x+ \sqrt{ a+ { x }^{ 2 } } })$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=4x\sqrt{x^{2}+1}+4x^{2}+1\text{, }&\left(arg(\sqrt{x^{2}+1}+2x)<\pi \text{ and }arg(x)\geq \pi \text{ and }x\neq 0\right)\text{ or }x=-\frac{\sqrt{3}}{3}\text{ or }\left(x\neq -\frac{\sqrt{2}}{4}\text{ and }arg(\sqrt{x^{2}+1}+2x)<\pi \right)\\a\neq 0\text{, }&x=0\end{matrix}\right.$$

Solve for a

$$\left\{\begin{matrix}a=4x\sqrt{x^{2}+1}+4x^{2}+1\text{, }&\left(4x\sqrt{x^{2}+1}+4x^{2}+1>0\text{ and }\sqrt{x^{2}+1}+2x\geq 0\right)\text{ or }\left(4x\sqrt{x^{2}+1}+5x^{2}+1\geq 0\text{ and }\sqrt{x^{2}+1}+2x\geq 0\text{ and }4x\sqrt{x^{2}+1}+4x^{2}+1\geq -x^{2}\text{ and }x<-\sqrt{-\left(4x\sqrt{x^{2}+1}+4x^{2}+1\right)}\text{ and }-\left(4x\sqrt{x^{2}+1}+4x^{2}+1\right)\geq 0\right)\text{ or }\left(4x\sqrt{x^{2}+1}+5x^{2}+1\geq 0\text{ and }4x\sqrt{x^{2}+1}+4x^{2}+1<0\text{ and }\sqrt{x^{2}+1}+2x\geq 0\text{ and }-\left(4x\sqrt{x^{2}+1}+4x^{2}+1\right)\geq 0\text{ and }4x\sqrt{x^{2}+1}+4x^{2}+1\geq -x^{2}\text{ and }x\leq -\sqrt{-\left(4x\sqrt{x^{2}+1}+4x^{2}+1\right)}\right)\text{ or }\left(4x\sqrt{x^{2}+1}+5x^{2}+1\geq 0\text{ and }x<0\text{ and }\sqrt{x^{2}+1}+2x\geq 0\text{ and }-\left(4x\sqrt{x^{2}+1}+4x^{2}+1\right)\geq 0\text{ and }x\leq -\sqrt{-\left(4x\sqrt{x^{2}+1}+4x^{2}+1\right)}\text{ and }4x\sqrt{x^{2}+1}+4x^{2}+1\geq -x^{2}\right)\\a>0\text{, }&x=0\end{matrix}\right.$$