Solving Logarithmic Equations

[lolily]

In my textbook, I saw an example for the problem log₃(x-4) - log₃ (x-1) = 1. The instructions simply say "Solve." The example solves the equation as normal, but then eventually arrives at x= -1/2, x=(4,∞), as the last step. But then the answer to the problem is no solution. Can anyone explain to me why this answer means that there's no solution? My textbook didn't really go over it, or I'm probably missing something basic. Thanks so much in advance!

 

[MarkFL]

The given equation implies:

[MATH]\frac{x-4}{x-1}=3[/MATH]
[MATH]x-4=3(x-1)=3x-3[/MATH]
[MATH]2x=-1[/MATH]
[MATH]x=-\frac{1}{2}\quad\checkmark[/MATH]
Now, what happens when we try to check this solution?