Your expression appears to be:

$\log^2 \frac{2}{3} + \log^2 1.5$

where $\log^2 x$ means $(\log x)^2$.

Step 1: Convert to Decimal Form

Since $\frac{2}{3} = 0.6667$ and $1.5 = \frac{3}{2}$, we rewrite the expression:

$(\log 0.6667)^2 + (\log 1.5)^2$

Step 2: Evaluate Logarithms

Using base 10 logarithms:

$\log 0.6667 \approx -0.1761$

$\log 1.5 \approx 0.1761$

Step 3: Square the Logarithms

$(-0.1761)^2 = 0.0310$

$(0.1761)^2 = 0.0310$

Step 4: Sum the Values

$0.0310 + 0.0310 = 0.0620$

Thus, the final result is:

$\mathbf{0.062}$

---

Would you like a more detailed explanation or a different base for the logarithm?

Here are some related questions you might find interesting:

1. What happens if we change the base of the logarithm to base 2?
2. How does the logarithm function behave for values between 0 and 1?
3. What is the general formula for squaring logarithms?
4. How does the sum of squared logarithms relate to logarithm properties?
5. Can we generalize this result for any two numbers $a$ and $b$?

Tip: Logarithm properties, like $\log(ab) = \log a + \log b$, can simplify complex expressions!