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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 + 0.8435Why did we do this?
 = \bar 2 \cdot 8435
Thus,
N = anti\log (0.8435) \times {10^{ - 2}}
6.974 \times {10^{ - 2}}
0.06974
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