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

To solve the equation:

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

we'll follow these steps:

### Step 1: Rewrite the logarithmic equation
Since the logarithm is base 10 (by convention unless otherwise specified), we can rewrite the equation in exponential form:

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

### Step 2: Simplify the exponential expression
The left-hand side can be simplified as follows:

\[
10^{\frac{3}{2}} = \left(\sqrt{10}\right)^3 = 10\sqrt{10}
\]

So the equation becomes:

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

### Step 3: Eliminate the cube root
To remove the cube root, we raise both sides to the power of 3:

\[
\left(10\sqrt{10}\right)^3 = w^2 - 10w
\]

### Step 4: Simplify the cubic expression
Calculate $\left(10\sqrt{10}\right)^3$:

\[
\left(10\sqrt{10}\right)^3 = 10^3 \times 10^{3/2} = 1000 \times 100\sqrt{10} = 100000\sqrt{10}
\]

So the equation now is:

\[
100000\sqrt{10} = w^2 - 10w
\]

### Step 5: Solve the quadratic equation
Rearrange the equation:

\[
w^2 - 10w - 100000\sqrt{10} = 0
\]

This is a quadratic equation of the form $ax^2 + bx + c = 0$ where $a = 1$, $b = -10$, and $c = -100000\sqrt{10}$.

### Step 6: Use the quadratic formula
The quadratic formula is:

\[
w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substituting the values:

\[
w = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(-100000\sqrt{10})}}{2(1)}
\]

\[
w = \frac{10 \pm \sqrt{100 + 400000\sqrt{10}}}{2}
\]

\[
w = \frac{10 \pm \sqrt{400000\sqrt{10} + 100}}{2}
\]

### Final Expression for $w$:
This expression for $w$ can be simplified further depending on the value, but in general:

\[
w = \frac{10 \pm \sqrt{400000\sqrt{10} + 100}}{2}
\]

This solution represents the roots of the equation. To find the exact value, further simplification or numerical computation may be required.

Would you like the numerical solution or more details on simplifying this expression?

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Here are some related questions for further exploration:

1. How do you simplify expressions involving radicals?
2. What are the properties of logarithms that can help solve logarithmic equations?
3. How do you convert between logarithmic and exponential forms?
4. What is the quadratic formula, and how is it derived?
5. What are the methods to solve quadratic equations besides using the quadratic formula?
6. How do you handle complex numbers if they appear in solutions?
7. What are the applications of logarithms in real-world problems?
8. How do you simplify expressions involving cube roots?

**Tip:** When solving logarithmic equations, always ensure that the arguments of the logarithms are positive, as the logarithm of a non-positive number is undefined in the real number system.

Learn how to solve the logarithmic equation (3/2) = log(sqrt[3](w^2 - 10w)) with a detailed step-by-step solution. This problem involves logarithms, exponential equations, and quadratic equations, suitable for advanced high school students (Grades 11-12).

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