<s>[INST] How to Solve : $ A =\frac{1}{6}\left((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3\right) $ $ A =\frac{1}{6}\left((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3\right).$
Solve for $2^A.$
(no calculators or graphs are permitted)
The way I went about solving this problem was using the properties of base $2$. For example, $2^x = 8$ , $x$ is obviously $3$, and $2^y = 16$, $y = 4 = x+1$, so adding $1$ to the exponent will give twice the answer. Since the arguments are twice each other, I used this pattern: 
$ A =\frac{1}{6}\left((x)^3-(x+1)^3-(x+2)^3+(x+3)^3\right)$
$= \frac{1}{6}\left((x^3-(x^3 + 3x^2 + 3x + 1)-(x^3 + 6x^2 + 12x + 8)+(x^3 + 9x^2 + 27x + 27)\right)$
Simplifying this we get $A = (12x + 18)/6 = 2x + 3.$
So now I solved for $2^{2x+3}$ using $2^{2x} =2^x * 2^x = 3^2 = 9.$ Now, since we add $3$ to the exponent, we have $2^{2x+3} = (2^3)(9) = 72.$
Would this be the correct answer, and if not, what would be a good way of going about it?
 [/INST] A good choice of substitution might help cut through the clutter of logs and nested parentheses, giving a visually pleasing and symmetrical solution.
Let $\text{L}^mn=(\log_2 n)^m$ and put $u=L3=\log_23$.
Hence 
$$\begin{align}
\text{L}^3 3&=(L3)^3&=u^3\\
\text{L}^3 6&=(L3+L2)^3&=(u+1)^3\\
\text{L}^3 12&=(L2+L4)^3&=(u+2)^3\\
\text{L}^3 24&=(L2+L8)^3&=(u+3)^3\end{align}$$
$$\begin{align}
A&=\frac16 ((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2 (24))^3)\\
&=\frac16 (L^33-L^36-L^312+L^324)\\
&=\frac 16 \left[u^3-(u+1)^3-(u+2)^3+(u+3)^3\right]\\
&=\frac 16 \left[ 12u+18\right]\\
&=2u+3\\
&=2L3+L8\\
&=L72=\log_2 72\\
2^A&=72\qquad \blacksquare 
\end{align}$$

Previous answer below:
$$\begin{align}\text{L}^3 3&=(\log_2 3)^3&&=u^3\\
\text{L}^3 6&=(\log_2 6)^3&=(\log_2 3+\log_2 2)^3&=(u+1)^3\\
\text{L}^3 12&=(\log_2 12)^3&=(\log_2 3+\log_2 4)^3&=(u+2)^3\\
\text{L}^3 24&=(\log_2 24)^3&=(\log_2 3+\log_2 8)^3&=(u+3)^3\end{align}$$
$$\begin{align}
A&=\frac16 ((\log_23)^3-(\log_2 6)^3-(\log_2 12)^3+(\log_2 24)^3)\\
&=\frac 16 \left[u^3-(u+1)^3-(u+2)^3+(u+3)^3\right]\\
&=\frac 16 \left[ 12u+18\right]\\
&=2u+3\\
&=2\log_2 3+\log_28\\
&=\log_2 72\\
2^A&=72\qquad \blacksquare 
\end{align}$$
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