http://www.boswell-beta.nl/docs/vbexwibeng.pdfExam Question 4, (a.)
I've been trying to solve the question above, but I'm having trouble understanding how to go about determining the x intercepts of the function, in particular, what to do with the natural log squared. I've tried looking for other explanations of problems similar to this but haven't found any so far. Any help would be much appreciated!
To find the domain, it's really simple... As $ln(x)$ is defined for $x \in ]0;+\infty[$, $(\ln x)^2$ will be defined over the same interval. so your solutions must belong to $\mathbb R_+^*$.
So let's solve it : you got $f(x) = (\ln x)^2 - 2\ln x$. $$f(x) = 0$$ $$\Rightarrow (\ln x)^2 - 2\ln x = 0$$ $$\Rightarrow (\ln x) (\ln x - 2) = 0$$ $$\Rightarrow \ln x = 0 \space\space OR\space\space \ln x -2 = 0$$ $$\Rightarrow x = e^0 \space\space OR \space\space \ln x = 2$$ $$\Rightarrow x = 1 \space\space OR \space\space x = e^2$$
1 and $e^2$ effectively belong to $\mathbb R_+^*$, so we can conclude : $$S = \{1;e^2\}$$