Page:Calculus Made Easy.pdf/175

Also, we see that $p$ is the numerical value of the ratio between the height of any ordinate and that of the next preceding it. In Fig. 40, we have taken $p$ as ${\tfrac {6}{5}}$; each ordinate being ${\tfrac {6}{5}}$ as high as the preceding one.

Fig. 40.	Fig. 41.

If two successive ordinates are related together thus in a constant ratio, their logarithms will have a constant difference; so that, if we should plot out a new curve, Fig. 41, with values of $\log _{\epsilon }y$ as ordinates, it would be a straight line sloping up by equal steps. In fact, it follows from the equation, that

${\begin{aligned}\log _{\epsilon }y&=\log _{\epsilon }b+x\cdot \log _{\epsilon }p,\\{\text{whence }}\quad \quad \quad \log _{\epsilon }y&-\log _{\epsilon }b=x\cdot \log _{\epsilon }p.\end{aligned}}$

Now, since $\log _{\epsilon }p$ is a mere number, and may be written as $\log _{\epsilon }p=a$, it follows that

$\log _{\epsilon }{\frac {y}{b}}=ax$,

and the equation takes the new form

$y=b\epsilon ^{ax}$.