evaluate without using a calculator

Question

[log(5)4 x log(2)10 ]/ [log(25)√10]

Answer 1

converting to base 10 logs, we have

(log4/log5 * 1/log2)/(log √10 / log25)
= (log4/(log5 log2)) * 2log5 / (1/2)
= log4 log10 log25 / log2 log5 log√10
= (4 log2 log5) / (1/2 log2 log5)
= 8