3^30 의 자릿수

먼저 ${\mathrm{\log}} \mathit{_{10}} 3 ^{30}$을 계산해 보면 ${\mathrm{\log}} \mathit{_{10}} 3 ^{30} = 30 {\mathrm{\log}} \mathit{_{10}} 3 = 30 \times 0.4771 = 14.313$이므로 $3 ^{30}$은 $15$자리의 수이다. $\therefore$$n = 15$ 한편, $14.3010 < 14.313 < 14.4771$이므로 $14 + 0.3010 < 14.313 < 14 + 0.4771$ ${\mathrm{\log}} \mathit{_{10}} 10 ^{14} + {\mathrm{\log}} \mathit{_{10}} 2 < {\mathrm{\log}} \mathit{_{10}} 3 ^{30} < {\mathrm{\log}} \mathit{_{10}} 10 ^{14} + {\mathrm{\log}} \mathit{_{10}} 3$ $\leftrightarrow$${\mathrm{\log}} \mathit{_{10}} \left( 2 \times 10 ^{14} \right) < {\mathrm{\log}} \mathit{_{10}} 3 ^{30} < {\mathrm{\log}} \mathit{_{10}} \left( 3 \times 10 ^{14} \right)$ $\therefore$$2 \times 10 ^{14} < 3 ^{30} < 3 \times 10 ^{14}$ 따라서, $3 ^{30}$의 가장 큰 자리의 숫자는 $2$이므로 $a = 2$ $\therefore$$n + a = 15 + 2 = 17$