Simplify $\ln(|x-x^2|) - \ln(|x-1|)$

Question

As the topic says, I need to simplify:

$$\ln |x-x^2| - \ln |x-1| $$

I don't know how to approach the problem at all. I'm not asking for the answer, but something to maybe get me going.

Answer 1

Remember that logarithms have some rules associated with them that help you simplify problems. The most common are: $\ln(ab) = \ln a + \ln b$ and $\ln(a/b) = \ln a - \ln b$. The second one is the key to your problem. If you have the difference of logs, say $\ln a - \ln b$ then you can simplify it to be the log of a quotient: $\ln(a/b)$. If you have a quotient of polynomials, then you should be trying to factorise and eliminate common factors. For example:

$$ \frac{x^2-x}{x-1} = \frac{x(x-1)}{x-1} = \ ? $$

You want to simplify $\ln |x^2-x| - \ln |x-1|,$ so apply the two steps that I suggest.

Answer 2

Hint: can you factor $x-x^2$ then use the laws of logs?