Question

How do you solve `log_10(x-1) - log_10(x+1) = 1`?

Answer 1

`x=-11/9`

Please see Comments by @Lotusbluete and @Tom below for conditions under which solution exists.
Mathematically stated: The solution exists only for the equation
`log_10|x-1| -log_10|x+1|=1`

Explanation:

On the RHS use the following Basic Log rule
`log_b(m/n) = log_b(m) – log_b(n)`
and on the LHS use the equality `log_10 10=1`
`log_10((x-1)/(x+1))=log_10 10`
Taking antilog of both sides

`(x-1)/(x+1)=10`
`=>(x-1)=10(x+1)`
`=>x-1=10x+10`
`=>9x=-11`
`=>x=-11/9`