1. Sketch, on the same set of axes, the curves

   $$y=2|\ln (x)|$$

   and

   $$y=\ln (x+1)$$

2. Find the value of $x>1$ for which these curves intersect.

3. By considering the other point of intersection, show that the cubic equation

   $$x^3+x^2-1=0$$

   has a real root between $0$ and $1$ (you do not need to calculate this root).

For (a), remember that $x\mapsto \ln (x)$ is the inverse function of $x\mapsto e^x$. To sketch an inverse function you need reflect in the line $y=x$.

Now, to get $y=\ln (x+1)$, consider this as a transformation of the curve $y=\ln (x)$.

The answer to (b) can be expressed in the form

where $a,b,c$ are coprime. Give the value of