Theory question 104
Logarithms(a) If Log102 = 0.3010 and log103 = 0.4771, calculate without using tables the value of Log 0.72. (b) A hawk on top of a tree, 20 metres high, views a chick on the ground at an angle of depression of 39°. Find correct to 2 significant figures the distance of the chick from the bottom of the tree.
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Model answer
(a) 0.72 = 72/100 = (23 × 32)/102.
log 0.72 = 3 log 2 + 2 log 3 − 2 = 3(0.3010) + 2(0.4771) − 2 = 0.9030 + 0.9542 − 2 = −0.1428 (or 1̄.8572).
(b) The angle of elevation of the hawk from the chick is also 39° (alternate angles).
tan 39° = 20/x, so x = 20/tan 39° = 20/0.8098 = 24.7 m.
Distance ≈ 25 m (2 s.f.).
Explanation
Write 0.72 using only powers of 2, 3 and 10, then use the laws of logarithms.