1. Find the characteristic of the following numbers
(i) \(5287\)
\[\text{Characteristic }= 3\]
(ii) \(59.28\)
\[\text{Characteristic }= 1\]
(iii) \( 0.0567\)
\[\text{Characteristic }=\overline{2}\]
(iv) \(234.7\)
\[\text{Characteristic }= 2\]
(v) \(0.000049\)
\[\text{Characteristic }=\overline{5}\]
(vi) \(145000\)
\[\text{Characteristic }= 5\]
2 Find the logarithm of the following numbers
(i) \(43\)
\[\text{Characteristic }= 1\]
\[\text{Mantissa }= 6335\]
\[\log(43) = 1.6335\]
(ii) \(579\)
\[\text{Characteristic }= 2\]
\[\text{Mantissa }= 7612\]
\[\log(579) = 2.7612\]
(iii) \(1.982\)
\[\text{Characteristic }= 0\]
\[\text{Mantissa }= 2967 + 4 = 2971\]
\[\log(1.982) = 0.2971\]
(iv) \(0.0876\)
\[\text{Characteristic }=\overline{2}\]
\[\text{Mantissa }= 9420\]
\[\log(0.0876) = \overline{2}.9420\]
(v) \(0.047\)
\[\text{Characteristic} = \overline{2}\]
\[\text{Mantissa} = 6721\]
\[\log(0.047) = \overline{2}.6721\]
(vi) \(0.000354\)
\[\text{Characteristic} = \overline{4}\]
\[\text{Mantissa} = 5478\]
\[\log(0.000354) = \overline{4}.5478\]
3. If \(\log 3.177 = 0.5019\), then find:
First, find Mantissa from \(3.177 = 0.5019\):
\[\text{Characteristic} = 0\]
\[\text{Mantissa} = 0.5019\]
(i) \(\log(3177)\)
\[\text{Characteristic} = 3\]
\[\log(3177) = 3.5019\]
(ii) \(\log(31.77)\)
\[\text{Characteristic} = 1\]
\[\log(31.77) = 1.5019\]
(iii) \(\log(0.03177)\)
\[\text{Characteristic} = \overline{2}\]
\[\log(0.03177) = \overline{2}.5019\]
4. Find the value of \(x\)
(i) \(\log x = 0.0065\)
\[x = \text{anti-log}(0.0065)\]
\[1014 + 1 = 1015\]
As the characteristic is \(0\), the answer is:
\[x = 1.015\]
(ii) \(\log x = 1.192\)
\[x = \text{anti-log}(1.192)\]
\[1556\]
As the characteristic is \(1\), the answer is:
\[x = 15.56\]
(v) \(\log(x) = 4.3561\)
\[x = \text{anti-log}(4.3561)\]
\[2270 + 1\]
\[2271\]
As the characteristic is \(4\), the answer is:
\[x = 22710\]
(iii) \(\log x = -3.434\)
\[\log x = -4 + 4 – 3.434\]
\[\log x = \overline{4} + 0.566\]
\[\log x = \overline{4}.566\]
\[x = \text{anti-log}(\overline{4}.566)\]
\[3681\]
As the characteristic is \(\overline{4}\), the answer is:
\[x = 0.0003681\]
(iv) \(\log x = -1.5726\)
\[\log x = -2 + 2 – 1.5726\]
\[\log x = \overline{2} + 0.4274\]
\[\log x = \overline{2}.4274\]
\[x = \text{anti-log}(\overline{2}.4274)\]
\[2673 + 2\]
\[ 2675\]
As the characteristic is \(\overline{2}\), the answer is:
\[x = 0.02675\]
(vi) \(\log x = -2.0184\)
\[\log x = -3 + 3 – 2.0184\]
\[\log x = \overline{3} + 0.9816\]
\[\log x = \overline{3}.9816\]
\[x = \text{anti-log}(\overline{3}.9816)\]
\[9572 + 13 \]
\[9585\]
As the characteristic is \(\overline{3}\), the answer is:
\[x = 0.009585\]
