This is a question about the application of the laws of logarithms.
Why isn't $\frac{1}{4}\log_2(8x - 56)^{16 }- 12 = \log_2((8x-56)^{16})^{\frac 1 4} - 12$ ?
According to the law of logarithms for powers, $\log_a p^n = n\log_a p$. I think I'm applying this rule correctly to the left member of the equation to obtain the right member. But when I evaluate the two expressions on my calculator, I get different results.
What am I doing wrong?
$\log_2 (8x-56)^{16}$ is ambiguous. Do you mean $(\log_2 (8x-56))^{16}$, or do you mean $\log_2\left( (8x-56)^{16}\right)$, i.e. is it the $16$'th power of the log, or is it the log of the $16$'th power? It makes a big difference! Your calculator is probably interpreting it as $(\log_2 (8x-56))^{16}$.