Solve analytically the logarithmic equation
$\displaystyle \log_{x+1}(\sqrt{2}-1)\cdot\log_{4-2\sqrt{2}}(x^2+2x+2)=1$
Let $\displaystyle a=\sqrt{2}-1$ and $\displaystyle b=x+1$. Then the equation becomes:
$\displaystyle \displaystyle{\frac{\ln{a}}{\ln{b}}\cdot \frac{\ln(b^2+1)}{\ln(a^2+1)}}=1$
or equivalently:
$\displaystyle \displaystyle{\frac{\ln{a}}{\ln(a^2+1)}=\frac{\ln b}{\ln(b^2+1)}}$
Now, the derivative of $\displaystyle \frac{\ln{t}}{\ln(t^2+1)}$ is
$\displaystyle \frac{\frac{\ln(t^2+1)}{t}-\frac{2t\ln t}{t^2+1}}{\ln^2(t^2+1)} = \frac{(t^2+1)\ln(t^2+1)-t^2\ln t^2}{t(t^2+1)\ln^2(t^2+1)} > 0$
So we conclude that the only solution is $\displaystyle a=b$, or:
$\displaystyle x=\sqrt{2}-2$