(a) Given that (log_{10} 2 = 0.3010, log_{10} 7 = 0.8451) and (log_{10} 5 = 0.6990), evaluate without using logarithm tables:
(i) (log_{10} 35); (ii) (log_{10} 2.8).
(b) Given that (N^{0.8942} = 2.8), use your result in (a)(ii) to find the value of N.
Explanation
(a)(i) (log_{10} 35 = log_{10} (7 times 5) = log_{10} 7 + log_{10} 5)
= (0.8451 + 0.6990 = 1.5441)
(ii) (log_{10} 2.8 = log_{10} (frac{4 times 7}{10} = log_{10} 4 + log_{10} 7 – log_{10} 10)
= (log_{10} 2^{2} + log_{10} 7 – 1)
= (2log_{10} 2 + log_{10} 7 – 1 = 2(0.3010) + 0.8451 – 1)
= (0.4471)
(b) (N^{0.8942} = 2.8)
(0.8942 log N = log 2.8)
(log N = frac{log 2.8}{0.8942} = frac{0.4471}{0.8942})
(log N = 0.5)