CAT 2020 Slot 3 · QA · Medium · Logarithms
2×4×8×6(log24)2(log48)3(log816)4\frac{2 \times 4 \times 8 \times 6}{(\log_2 4)^2 (\log_4 8)^3 (\log_8 16)^4}(log24)2(log48)3(log816)42×4×8×6 equals
24
Numerator = 2×4×8×6=3842 \times 4 \times 8 \times 6 = 3842×4×8×6=384.
Denominator terms: log24=2 ⟹ (log24)2=4\log_2 4 = 2 \implies (\log_2 4)^2 = 4log24=2⟹(log24)2=4. log48=32 ⟹ (log48)3=278\log_4 8 = \frac{3}{2} \implies (\log_4 8)^3 = \frac{27}{8}log48=23⟹(log48)3=827. log816=43 ⟹ (log816)4=25681\log_8 16 = \frac{4}{3} \implies (\log_8 16)^4 = \frac{256}{81}log816=34⟹(log816)4=81256.
Denominator = 4×278×25681=4×13×323=1283×33…4 \times \frac{27}{8} \times \frac{256}{81} = 4 \times \frac{1}{3} \times \frac{32}{3} = \frac{128}{3} \times \frac{3}{3} \dots4×827×81256=4×31×332=3128×33… Wait: 4×278×25681=12×27×25681=2566=164 \times \frac{27}{8} \times \frac{256}{81} = \frac{1}{2} \times 27 \times \frac{256}{81} = \frac{256}{6} = 164×827×81256=21×27×81256=6256=16.
Fraction = 38416=24\frac{384}{16} = 2416384=24.
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