TITLE: If $3^x = 5$, $5^y = 10$, $10^z = 16$, then what is $3^{xyz}$?
QUESTION [5 upvotes]: Can't post images so I'll type it here:
$$3^x = 5,\qquad 5^y = 10,\qquad 10^z = 16$$
Then what is $3^{xyz}$?
I've spent like an hour trying to solve it and I failed. Help would be super duper appreciated.
Thank you!
Edit: uhh I think I solved it? Would the answer be $16$?
Basically I put $3^x$ in place of the $5$ in $5^y = 10$, so now I have $(3^x)^y = 10$ (which is $3^{xy} = 10$), did the same for the last equation and I got $16$ as an answer, but can anyone confirm this?

REPLY [6 votes]: Ricky gives the best solution but here is a brute force solution that uses more machinery. We note that
$$
x=\frac{\log 5}{\log 3};\quad y=\frac{\log 10}{\log 5}; \quad z=\frac{\log 16}{\log 10}
$$ 
so
$$
xyz=\frac{\log 16}{\log 3}=\log_{3}16
$$
whence
$$
3^{xyz}=16.
$$