The given expression is:

$\ ^2 \log 8 + ^2 \log 6 - ^2 \log 3$

This notation suggests that we are dealing with logarithms of base 2. Thus, it can be written as:

$\log_2 8 + \log_2 6 - \log_2 3$

Step-by-step solution:

1. Simplify each term:

   • $\log_2 8$: Since $8 = 2^3$, we have $\log_2 8 = 3$.
   • $\log_2 6$: We can leave $\log_2 6$ as is because it cannot be simplified further using powers of 2.
   • $\log_2 3$: Similarly, $\log_2 3$ stays as is.

2. Apply logarithmic properties: Using the property $\log_b (A) + \log_b (B) = \log_b (A \cdot B)$, we combine the first two terms:

   $\log_2 (8 \cdot 6) = \log_2 48$

   Now the expression becomes:

   $\log_2 48 - \log_2 3$

3. Simplify further: Using another property $\log_b (A) - \log_b (B) = \log_b \left( \frac{A}{B} \right)$, we can combine the remaining terms:

   $\log_2 \left( \frac{48}{3} \right) = \log_2 16$

4. Final result: Since $16 = 2^4$, we have:

   $\log_2 16 = 4$

Thus, the value of the expression is 4.

Would you like further details or explanations on logarithmic properties? Here are some related questions to consider:

1. How do you change the base of a logarithm?
2. What are some common logarithmic identities?
3. Can logarithms be used to solve exponential equations?
4. What is the graphical interpretation of logarithmic functions?
5. How does the logarithm function behave for values between 0 and 1?

Tip: When simplifying logarithmic expressions, always look for opportunities to apply the properties of logarithms, such as product, quotient, and power rules.