[Math] Simplify $(\log 2)^3+(\log 5)^3+(\log 2)(\log 125)$

Question

Can't seem to figure out how to tackle this one. I know $\log 5 = 1 – \log 2$, but I don't see a way to get around the cubed logarithms except for brute force. The answer is $1$. Using the sum of cubes formula gets me
$$
(\log 2 + 1 – \log 2)[(\log 2)^2 – \log 2(1 – \log 2) + (1 – \log 2)^2] \;,
$$ which is so complex that I feel the question isn't meant to be solved this way. Any input would be appreciated. 🙂

Answer 1

$(\log2)^3+(\log5)^3+(\log2)(\log125)\\=(\log2)^3+(\log5)^3+3(\log2)(\log5)(\log2+\log5)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \because \log2+\log5=\log10=1\\=(\log2+\log5)^3=1$