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TMUA 2017 · Paper 2 · Question 8 of 20

TMUA 2017 Paper 2 Question 8

Exponentials and logarithms — Comparing logs, powers and trigonometric values. Try it first; the answer and a full worked solution are below.

TMUA 2017 · Paper 2Exponentials and logarithmsComparing logs, powers and trigonometric values5 options
Which one of the following numbers is smallest in value?
  1. Alog2 7
  2. B(2−3 + 2−2)−1
  3. C2(π/3)
  4. D14(√2 − 1)3
  5. E4sin2(π4)
Show the answer and worked solution
answer · E
  1. Alog2 7
  2. B(2−3 + 2−2)−1
  3. C2(π/3)
  4. D14(√2 − 1)3
  5. E4sin2(π4)
Get each into a form you can compare with 2 and 3. log2 7 sits between log2 4 = 2 and log2 8 = 3, and is close to 2.8. The second is (18 + 14)−1 = 83 ≈ 2.67. Since π3 ≈ 1.05, the third is a little above 2. The fourth has √2 − 1 ≈ 0.414, so its cube is about 0.071 and the whole thing is about 3.5. The last is 4(1√2)2 = 2 exactly, which is the smallest.