To solve the given equation
[tex]\[
\frac{2}{3} = \log\left( \sqrt[3]{w^2 - 10w} \right),
\][/tex]
we will follow a step-by-step algebraic approach.
### Step 1: Rewrite the equation in exponential form
Recall that the logarithmic form [tex]\( a = \log_b c \)[/tex] can be rewritten as [tex]\( b^a = c \)[/tex]. Since the base of the logarithm is not provided explicitly, it is typically assumed to be 10.
Thus,
[tex]\[
\frac{2}{3} = \log\left( \sqrt[3]{w^2 - 10w} \right)
\][/tex]
is equivalent to
[tex]\[
10^{\frac{2}{3}} = \sqrt[3]{w^2 - 10w}.
\][/tex]
### Step 2: Remove the cube root by cubing both sides of the equation
Cubing both sides to eliminate the cube root gives:
[tex]\[
\left(10^{\frac{2}{3}}\right)^3 = \left( \sqrt[3]{w^2 - 10w} \right)^3.
\][/tex]
Simplifying both sides, we get:
[tex]\[
10^2 = w^2 - 10w.
\][/tex]
So, we have:
[tex]\[
100 = w^2 - 10w.
\][/tex]
### Step 3: Rearrange into a standard quadratic equation form
Rearrange the equation to look like a standard quadratic equation:
[tex]\[
w^2 - 10w - 100 = 0.
\][/tex]
### Step 4: Solve the quadratic equation using the quadratic formula
Recall the quadratic formula, [tex]\( w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where in our equation [tex]\( a = 1 \)[/tex], [tex]\( b = -10 \)[/tex], and [tex]\( c = -100 \)[/tex].
Substituting these values in, we get:
[tex]\[
w = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot (-100)}}{2 \cdot 1}
\][/tex]
[tex]\[
w = \frac{10 \pm \sqrt{100 + 400}}{2}
\][/tex]
[tex]\[
w = \frac{10 \pm \sqrt{500}}{2}
\][/tex]
[tex]\[
w = \frac{10 \pm 10\sqrt{5}}{2}
\][/tex]
[tex]\[
w = 5 \pm 5\sqrt{5}.
\][/tex]
### Step 5: Simplify the solutions
So, the simplified solutions are:
[tex]\[
w = 5 + 5\sqrt{5} \quad \text{and} \quad w = 5 - 5\sqrt{5}.
\][/tex]
Examining these values numerically,
[tex]\[
5 + 5\sqrt{5} \approx 16.1803398874989
\][/tex]
[tex]\[
5 - 5\sqrt{5} \approx -6.18033988749895.
\][/tex]
### Conclusion
Thus, the solutions to the equation [tex]\(\frac{2}{3} = \log(\sqrt[3]{w^2 - 10w})\)[/tex] are approximately:
[tex]\[
w \approx -6.18033988749895 \quad \text{and} \quad w \approx 16.1803398874989.
\][/tex]
