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146

Calculus Made Easy

The points calculated are:

For Fig. 38 ${\begin{cases}\\\\\\\end{cases}}$	$\,\,\,x\,\,\,$	$\,\,\,0\,\,\,$	$0.5$	$1$	$1.5$	$2$
For Fig. 38 ${\begin{cases}\\\\\\\end{cases}}$	$y$	$1$	$1.65$	$2.71$	$4.50$	$7.69$

For Fig. 39 ${\begin{cases}\\\\\\\end{cases}}$	$\,\,\,x\,\,\,$	$\,\,\,1\,\,\,$	$2$	$3$	$4$	$8$
For Fig. 39 ${\begin{cases}\\\\\\\end{cases}}$	$y$	$0$	$0.69$	$1.10$	$1.39$	$2.08$


Fig. 38.	Fig. 39.

It will be seen that, though the calculations yield different points for plotting, yet the result is identical. The two equations really mean the same thing.

As many persons who use ordinary logarithms, which are calculated to base $10$ instead of base $\epsilon$ , are unfamiliar with the “natural” logarithms, it may be worth while to say a word about them. The ordinary rule that adding logarithms gives the logarithm of the product still holds good; or

$\log _{\epsilon }a+\log _{\epsilon }b=\log _{\epsilon }ab$ .

Also the rule of powers holds good;

$n\times \log _{\epsilon }a=\log _{\epsilon }a^{n}$ .