ONE MORE STEP TOWARD BETTER TOMORROW : chapter 3

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,

	$N = anti\log (0.8435) \times {10^{ - 2}}$
	$6.974 \times {10^{ - 2}}$
	$0.06974$

Saturday, 9 August 2014

chapter 3 - Worked Out Examples

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