The equation is:

\[
x^3 \log x = 100x
\]

To solve this, we can approach it step by step:

### Step 1: Divide both sides by \(x\)
As long as \(x \neq 0\), we can divide both sides of the equation by \(x\):

\[
x^2 \log x = 100
\]

### Step 2: Isolate \(\log x\)
Now, isolate \(\log x\) by dividing both sides by \(x^2\):

\[
\log x = \frac{100}{x^2}
\]

### Step 3: Solve using numerical methods
This is a transcendental equation and doesn’t have an algebraic solution. We can solve it numerically to find the value of \(x\) by plotting or using numerical approximation techniques like Newton's method.

Would you like me to calculate an approximate numerical solution for \(x\)? 

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Here are 5 related questions for practice:

1. What is the general approach to solving transcendental equations?
2. How does the logarithmic function affect polynomial equations?
3. Can \(x = 0\) ever be a solution to the equation \(x^3 \log x = 100x\)?
4. What are the common methods for solving non-linear equations numerically?
5. How do you apply Newton’s method to an equation like this?

**Tip**: For transcendental equations, graphical solutions and numerical methods are often the most effective approach!

Learn how to solve the transcendental equation x^3 log x = 100x step-by-step. This solution covers key mathematical concepts such as logarithmic functions, polynomial equations, and the use of numerical methods like Newton's method. Suitable for advanced mathematics students.

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