From the graph, we can visually approximate the point where the functions $f(x) = (\ln x)^2$ and $g(x) = 7x$ intersect. It looks like the intersection occurs around $x \approx 0.26$.

Let's now calculate the corresponding $y$-coordinate at this point, and verify it numerically.

I will refine this approximation.The point of intersection between the functions $f(x) = (\ln x)^2$ and $g(x) = 7x$ is approximately:

$(x, y) \approx (0.26, 1.82)$

Would you like further details on this process or any additional help with this concept?

Here are 5 related questions to further your understanding:

1. What are other methods for finding intersections besides graphical approximation?
2. How does the natural logarithm function behave for small and large values of $x$?
3. How can numerical methods, like the Newton-Raphson method, refine approximations?
4. What is the derivative of $f(x) = (\ln x)^2$, and how can it be used to study the function's behavior?
5. Can there be multiple intersections between two functions? How would we determine that?

Tip: Always start with a reasonable guess when using numerical solvers like fsolve, based on a graphical or logical approximation.