Question

What are the restrictions on the variable x in the following equation? Why do these restrictions exist?

log(x-2)-log(3x-5)=5

Thanks for your help :-)

Answer 1

log(n) is defined only for n > 0.

So, we must have x-2 > 0 or x>2
and 3x-5 > 0 or x > 5/3

Combining those conditions, we see we must have x>2. That will make both expressions positive.

Answer 2

Since we cannot take Log of zero, x-2
cannot equal 0. In other words, x cannot equal 2. So x can be any real
number except 2:

X < > 2.

Also, 3x - 5 cannot equal 0:

3x-5 = 0,
3x = 5,
X = 5/3.
So x should be less than or greater than 5/3:

X < > 5/3.