Area Between $\log(x)$ And $(\log(x))^2$

Question

This is a homework question. Find the area of the figure bounded by the curves $y=\log(x)$ and $y=(\log x)^2$ . I tried to plot a rough graph of the given curves and found that the graph of the first function came from $-\infty$ to $x=1$ and rose up in the first quadrant thereafter . Similarly for the second curve , it's graph descend from $+\infty$ to $x=1$ and after meeting the first curve at the point $(e,1)$ from below rose up in the first quadrant always remaining upside $y=\log(x)$. I found the 3 different areas and added them up and got the answer as $6-e$ sq. units but according to my book the answer is $3-e$ sq. units. Any help would be highly appreciated .

Thanks.

Answer 1

Don't include the area between x=0 to x=1 that is the area bound by the two curves with x=0. Therefore the remaining area from x=1 to x=e is 3-e.

Answer 2

$\int_0^1\left( \log(x)^2 - \log(x)\right) dx = 3$

$\int_1^e \left(\log(x)- \log(x)^2\right) dx = 3 - e$