The image contains a logarithmic problem involving simplifications. Here's a step-by-step breakdown of the parts visible:

Equation:

The main expression involves several logarithmic terms. It seems to involve: $3 \log 24 - 3 \log (2 \sqrt{3}) + 3 \log \left(\frac{81}{9}\right)$

Step-by-Step Breakdown:

1. First term: $3 \log 24$ This can remain as it is for now, though further simplification could involve breaking 24 down into prime factors.

2. Second term: $-3 \log (2 \sqrt{3})$ Using logarithmic properties, we can split the logarithm: $3 \log (2 \sqrt{3}) = 3 \log 2 + 3 \log \sqrt{3} = 3 \log 2 + \frac{3}{2} \log 3$

3. Third term: $3 \log \left(\frac{81}{9}\right)$ This simplifies using the property $\log \left(\frac{a}{b}\right) = \log a - \log b$: $3 \log \left(\frac{81}{9}\right) = 3 (\log 81 - \log 9)$ $\log 81 = \log (3^4) = 4 \log 3 \quad \text{and} \quad \log 9 = \log (3^2) = 2 \log 3$ So, we have: $3 (4 \log 3 - 2 \log 3) = 3 (2 \log 3) = 6 \log 3$

Combined expression:

Now, combining all terms: $3 \log 24 - (3 \log 2 + \frac{3}{2} \log 3) + 6 \log 3$ This simplifies further as: $3 \log 24 - 3 \log 2 - \frac{3}{2} \log 3 + 6 \log 3$ $= 3 (\log 24 - \log 2) + (6 \log 3 - \frac{3}{2} \log 3)$ $= 3 \log 12 + \left(\frac{9}{2} \log 3\right)$

The final simplified form is: $3 \log 12 + \frac{9}{2} \log 3$

Let me know if you'd like any specific details explained or any further breakdown!

Related Questions:

1. What are the logarithmic properties used in this problem?
2. How does factorization help simplify logarithmic expressions?
3. What are common mistakes when simplifying logarithmic terms?
4. How do we apply the change of base formula to logarithms?
5. How does one expand or condense logarithmic expressions?

Tip: Always use the properties of logarithms to break down complex terms before solving or simplifying.