$$ \frac32 $$$$ =\log^3 $$$$ \sqrt{w^2-10w} $$

The equation you've written is:

\[
\frac{3}{2} = \log^3 \sqrt{w^2 - 10w}
\]

To solve for $ w $, let's follow these steps:

1. **Simplify the logarithmic expression**:  
   \[
   \frac{3}{2} = \left(\log \sqrt{w^2 - 10w}\right)^3
   \]
   Since $\log \sqrt{w^2 - 10w}$ is the cube root, take the cube root of both sides to simplify:
   \[
   \left(\frac{3}{2}\right)^{1/3} = \log \sqrt{w^2 - 10w}
   \]

2. **Simplify further**:  
   \[
   \left(\frac{3}{2}\right)^{1/3} = \frac{1}{2} \log (w^2 - 10w)
   \]
   Multiply both sides by 2:
   \[
   2\left(\frac{3}{2}\right)^{1/3} = \log (w^2 - 10w)
   \]

3. **Exponentiate both sides**:  
   Since $ \log (w^2 - 10w) = x $ implies $ w^2 - 10w = 10^x $, we can write:
   \[
   w^2 - 10w = 10^{2\left(\frac{3}{2}\right)^{1/3}}
   \]

4. **Solve the quadratic equation**:  
   Let's solve for $ w $ using the quadratic formula:
   \[
   w = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(-10^{2\left(\frac{3}{2}\right)^{1/3}})}}{2(1)}
   \]
   Simplifying further will provide the possible values of $ w $.

Let me know if you would like the detailed calculation of this quadratic equation or have any specific questions!

---

**Would you like to dive into the details of the calculation?**

Here are some related questions that you might find useful:

1. What is the cube root of a number?
2. How do logarithms with different bases work?
3. What are the properties of logarithmic functions?
4. How can the quadratic formula be applied to solve for $ w $?
5. How do you convert between logarithmic and exponential forms?
6. What is the significance of the discriminant in the quadratic equation?
7. What happens when $ w^2 - 10w $ is negative?
8. How do logarithmic identities simplify expressions?

**Tip:** Understanding the properties of logarithms can help simplify and solve more complex equations.

Learn how to solve the logarithmic equation 3/2 = log^3 sqrt(w^2 - 10w) with a detailed step-by-step solution. This problem covers key concepts in logarithms, cube roots, and quadratic equations. Suitable for advanced high school students.

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