1/4 log^3 16/81-1/3 log^3 8/27

Let's simplify the expression \( \frac{1}{4} \log^3 \frac{16}{81} - \frac{1}{3} \log^3 \frac{8}{27} \).

First, we can rewrite the logarithms using their properties:

1. \( \frac{16}{81} = \frac{2^4}{3^4} = \left(\frac{2}{3}\right)^4 \)
2. \( \frac{8}{27} = \frac{2^3}{3^3} = \left(\frac{2}{3}\right)^3 \)

Now we can rewrite the logarithmic expressions:

\[ \log \frac{16}{81} = \log \left(\left(\frac{2}{3}\right)^4\right) = 4 \log \frac{2}{3} \]

\[ \log \frac{8}{27} = \log \left(\left(\frac{2}{3}\right)^3\right) = 3 \log \frac{2}{3} \]

Now substituting back into our expression:

\[ \frac{1}{4} \log^3 \frac{16}{81} = \frac{1}{4} (4 \log \frac{2}{3})^3 = \frac{1}{4} \cdot 64 \log^3 \frac{2}{3} = 16 \log^3 \frac{2}{3} \]

\[ \frac{1}{3} \log^3 \frac{8}{27} = \frac{1}{3} (3 \log \frac{2}{3})^3 = \frac{1}{3} \cdot 27 \log^3 \frac{2}{3} = 9 \log^3 \frac{2}{3} \]

Putting this all together:

\[ 16 \log^3 \frac{2}{3} - 9 \log^3 \frac{2}{3} = (16 - 9) \log^3 \frac{2}{3} = 7 \log^3 \frac{2}{3} \]

Thus, the final simplified expression is:

\[ \boxed{7 \log^3 \frac{2}{3}} \]