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How to use this applet

If the two points are on opposite sides of the

In the equation *y*=*Ce ^{kx}*, the two constants

Now solve each of these equations for

Since both equations represent

Now solve this equation for

Now take natural logarithms on both sides:

This is where it is important that the two points
(*x*_{0},*y*_{0}) and
(*x*_{1},*y*_{1}) be on the same side of
the *x*-axis -- in other words, that *y*_{0}
and *y*_{1} have the same sign, so that
*y*_{1}/*y*_{0} is positive and the logarithm
on the left hand side is defined. If *y*_{0}
and *y*_{1} have the same sign (which can be checked with
the inequality *y*_{0}*y*_{1}>0)
then
ln(*y*_{1}/*y*_{0})
=ln(|*y*_{1}|)-ln(|*y*_{0}|) by properties of
logarithms, with the absolute values guaranteeing that the logarithms
will be defined even if *y*_{0} and *y*_{1}
are negative. Some care must be taken at this point
to check the inequality *y*_{0}*y*_{1}>0,
though, since this expression with the absolute values no longer requires
*y*_{0} and *y*_{1} to have the same sign.

Continuing to solve for *k*, the equation now looks like:

Solving this algebraically for

It is interesting to notice that the above equation resembles the slope of a line.

Now that the value of *k* is known (from the points
(*x*_{0},*y*_{0}) and
(*x*_{1},*y*_{1})), finding *C* is
easier: plug either point into the equation
*y*=*Ce ^{kx}* -- plugging in
(

These formulas for