let,
$\log_{17}275 =x$
so,$17^{x}=275$ …………….(1)
$\log_{19}375=y$
so,$19^{y}=375$ …………………...(2)
(1)$\div$(2)
$\frac{17^{x}}{19^{y}}=\frac{275}{375}< 1$
so,$17^{x}<19 ^{y}$
taking log both sides respect to base 19,
$x*\log_{19}17< y$
so, $\frac{x}{y}< 1$ ………..[as $\log_{17}19 > 1$]
so,$x< y$
[note ,
$\log_{17}275=1.984$
$\log_{19}375=2.012$
if you are allowed to use calculator which i believe you are not]