## Saturday, 9 August 2014

### chapter 3 - Worked Out Examples

 Example: 1
 Use logs to evaluate $N = \dfrac{{647 \cdot 32 \times 0.00000147}}{{8.473 \times 64}}$
 Solution: 1
Our approach consists of four steps:
• – convert the expression for $N$ into logs
• evaluate those logs using a logs – table
• thus determine $\log N$
• calculate antilog of $\log N$
Thus,
 $\log N = \left( {\dfrac{{647 \cdot 32 \times 0.00000147}}{{8.473 \times 64}}} \right)$ $= \log \left( {647 \cdot 32} \right) + \log \left( {0.00000147} \right) - \log \left( {8.473} \right) - \log \left( {64} \right)$
Now, note that (use log tables):
 $647 \cdot 32 = 6.4732 \times {10^2}$ $\Rightarrow \,\,\,\, \log \left( {647 \cdot 32} \right) = 2 + \log \left( {6.4732} \right)$ $= 2 + \,\,\, \cdot 8111$ $= 2 \cdot 8111$
 $0.00000147 = 1 \cdot 47 \times {10^{ - 6}}$ $\Rightarrow \,\,\,\, \log \left( {0 \cdot 00000147} \right) = - 6 + \log \left( {1 \cdot 47} \right)$ $= \bar 6 \cdot 1673$ $8.473 = 8.473 \times {10^0}$ $\Rightarrow \log \left( {8.473} \right) = 0 + 0.9820$ $= 0.9820$
 $64 = 6.4 \times {10^1}$ $\Rightarrow \log \left( {64} \right) = 1 + \log (6.4)$ $= 1.8062$
Thus,
 $\log N = 2.8111 + \bar 6.1673 - 0.9280 - 1.8062$ $= 2.8111 + ( - 6 + 0.1673) - 0.9280 - 1.8062$ $\left. \begin{array}{l} = - 5.7558\\ = - 5 - 0.7558\\ = \left( { - 5 - 1} \right) + 1 - 0.7558\\ = - 6 + 0.2442\\ = \bar 6 \cdot 2442 \end{array} \right\}$ $\begin{array}{l} {\rm{Conversion}}\,{\rm{to \,standard}}\\ \left( {{\rm{Characteristic}}} \right)\,.\left( {{\rm{Mantissa}}}\, \right)\\ {\rm{form}}\,{\rm{.}}\\ {\rm{Note\, that\, according\, to\, our \,convention,\,}}\\ {\rm{the \,mantissa\,}}\\ {\rm{must\, be\, positive\,}}{\rm{.}} \end{array}$
Now, we have the characteristic and mantissa of $\log N$.
We thus find antilog (mantissa) $= anti\log (0.2442) = 1.7547$ {from your antilog tables}
 $\Rightarrow N = 1.7547 \times {10^{ - 6}}$ $= 0.00000\,17547$
Verify this using a calculator. Note that $N$ was calculated without any actual multiplication and division – operations much more cumbersome than addition or subtraction.
 Example: 2
 Find the value of $\sqrt[5]{{0.00000165}}$
 Solution: 2
Let
 $N = \sqrt[5]{{0.00000165}}$ $\Rightarrow \,\,\,\, \log \,N = \dfrac{1}{5}\log \left( {0.00000165} \right)$ $= \dfrac{1}{5}\left( { - 6 + \log 1.65} \right)$ (Why?) $= \dfrac{1}{5}\left( { - 6 + .2175} \right)$ $= - 1.2 + 0.0435$ $= - 2 + 0.8435$ Why did we do this? $= \bar 2 \cdot 8435$
Thus,
 $N = anti\log (0.8435) \times {10^{ - 2}}$ $6.974 \times {10^{ - 2}}$ $0.06974$