To solve the equation \( x \log(\log(32)) = \frac{2}{3} \), let's go step by step:

### Step 1: Simplify the logarithmic expression
First, we need to find \( \log(32) \). Assuming the logarithm is in base 10:

\[
\log(32) \approx 1.5051
\]

### Step 2: Take the logarithm of the result
Now, let's compute \( \log(\log(32)) \):

\[
\log(1.5051) \approx 0.1775
\]

### Step 3: Substitute back into the equation
Now, substitute \( \log(\log(32)) \approx 0.1775 \) into the original equation:

\[
x \cdot 0.1775 = \frac{2}{3}
\]

### Step 4: Solve for \( x \)
To solve for \( x \), divide both sides by \( 0.1775 \):

\[
x = \frac{\frac{2}{3}}{0.1775} = \frac{2}{3} \times \frac{1}{0.1775} = \frac{2}{3} \times 5.6338
\]

\[
x \approx 3.7559
\]

### Final Answer:
\[
x \approx 3.76
\]

Would you like further details or have any questions?

### Related questions:
1. How do you calculate logarithms in different bases?
2. Can you explain the properties of logarithms?
3. What is the difference between natural logarithms and common logarithms?
4. How can we approximate logarithms without a calculator?
5. How are logarithms used in solving exponential equations?

### Tip:
When solving equations involving logarithms, always make sure to confirm the base of the logarithm, as different bases (e.g., base 10 vs. natural logarithms) can change the outcome.

Step-by-step solution to the logarithmic equation x log log 32 = 2/3, involving logarithmic properties and algebraic methods. Includes detailed calculations for log(32) and solving for x. Suitable for high school students (Grades 10-12).

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