Вычислите: log2(10)*log3(2)* log4(3)* log5(4)*...*log1000(999)

$log_{2}10*log_{3}2*log_{4}3*log_{5}4*...*log_{1000}999$

перейдем к основанию 2:

$log_{2}10*\frac{log_{2}2}{log_{2}3}*\frac{log_{2}3}{log_{2}4}*\frac{log_{2}4}{log_{2}5}*...*\frac{log_{2}999}{log_{2}1000}=\\ =\frac{log_{2}10}{log_{2}1000}=\\ = \frac{log_{2}10}{log_{2}10^{3}}=\\ =\frac{log_{2}10}{3log_{2}10}=\frac{1}{3}$