Problem Analysis #common_06

Recorded Python call:

{}

Recorded Python call:

from fractions import Fraction
from sympy import symbols, log, Rational, solveset, S, Eq

# a, b > 1 with log_a b = 3  ->  b = a^3
# log_3 (b/a) = 1/2        ->  b/a = 3^(1/2)
a = symbols('a', positive=True)
b = a**3
# condition: b/a = sqrt(3)  => a^2 = sqrt(3) => a = 3^(1/4)
sol = solveset(Eq(a**2, 3**Rational(1,2)), a, domain=S.Reals)
print("candidate a:", sol)
a_val = list(sol)[0]
b_val = a_val**3
print("check a>1,b>1:", a_val > 1, b_val > 1)
print("check log_a b = 3:", log(b_val, a_val).simplify())
print("check log_3(b/a) = 1/2:", log(b_val/a_val, 3).simplify())
ab = a_val * b_val          # = a^4 = 3
print("ab simplifies to:", ab.simplify(), "== 3:", ab == 3)
print("log_9(ab) =", log(ab, 9).simplify())
print("as fraction:", Fraction(1,2))

풀이 (계산으로 확인):

• $\log_a b = 3 \Rightarrow b = a^3$ (a,b>1)
• $\log_3 \frac ba = \frac12 \Rightarrow \frac ba = 3^{1/2}$

두 식을 결합하면 $b/a = a^2 = 3^{1/2}$이므로 $a = 3^{1/4}$, $b = 3^{3/4}$ (둘 다 1보다 큼, 위 실행에서 조건 $\log_a b=3$, $\log_3(b/a)=\frac12$ 만족 확인).

따라서 $ab = 3^{1/4}\cdot 3^{3/4} = 3$이고,

Python 심볼릭 계산에서도 $\log_9(ab) = 1/2$가 확인되었습니다. 이는 선택지 ②에 해당합니다.

Recorded Python call:

from sympy import Rational
print("choice 2 corresponds to:", Rational(1,2))

$\boxed{2}$