Hi everyone : hope ur well so I have this log plus absolute question so. For ln (x-2) = ln (2-x) I know there is no solution however for ln |x-2| = ln |2-x| There is , someone can kindly how is that possible and also help me solve this thank u

Question

Hi everyone : hope ur well so I have this log plus absolute question so. For ln (x-2) = ln (2-x) I know there is no solution however for  ln |x-2| = ln |2-x| There is , someone can kindly how is that possible and also help me solve this thank u

perl · Answer

I don't think there is a solution in either case.

anonymous · Answer

$$\ln|x-2|=\ln\left(|-1||2-x|\right)=\ln|2-x|$$

anonymous · Answer

You can think of the absolute value terms $|2-x|$ and $|x-2|$ in terms of distance. The distance between $x$ and $2$ is always the same.

$|a-b|$ can be read as "the distance from $a$ to $b$", and $|b-a|$ as "the distance from $b$ to $a$". You can expect the distance to be the same regardless of the starting point.