SE-SKOLORNAS 1972 Problem 4

Put $x = \log_{10} 2$ , $y = \log_{10} 3$ . Then $15 < 16$ implies $1 - x + y < 4x$ , so $1 + y < 5x$ . Derive similar inequalities from $80 < 81$ and $243 < 250$ . Hence show that $$0.47 < \log_{10} 3 < 0.482.$$