The equation provided in the image is:

$x^2 \log{x} + 3.5 = 100 \sqrt{x}$

To solve this equation for $x$, here’s the step-by-step process:

Step 1: Isolate terms involving $x$

Subtract $100 \sqrt{x}$ from both sides:

$x^2 \log{x} + 3.5 - 100 \sqrt{x} = 0$

At this point, the equation is non-linear and will likely require numerical methods to find the exact solution. A possible approach would be to either approximate values for $x$ using trial and error or apply a numerical solver to obtain the solution.

Would you like me to proceed with solving this equation numerically?

Meanwhile, here are some related questions to expand your understanding:

1. What are the properties of logarithmic functions that are useful in solving equations?
2. How can you simplify expressions involving square roots and powers of $x$?
3. What is the difference between solving algebraic equations and transcendental equations like this one?
4. How does the change of base formula for logarithms work?
5. What methods can be used to approximate solutions for non-linear equations like this?

Tip: When dealing with complex equations, graphing both sides of the equation can help you visualize the solutions!