How can I find slope and y-intercept of the equation y=ab^x by taking the natural log of each side?

Answer 1

ln y = ln a + x ln b

This shows that if you plot ln y as a function of x, you will get a straight line with x intercept of ln a and slope of ln b.

Answer 2

y = ab^x

Take the natural log, or ln, of both sides.

Recall that there are only three rules that apply to logarithms:

ln(ab) = lna + lnb

ln(a/b) = lna - lnb

ln(a^x) = xlna

where x, a, b are real.

lny = ln[ab^x]

lny = lna + ln(b^x)

lny = lna + xlnb (*)

If we want the y-intercept, let x = 0, then lny = lna, or y = a. So, the y-intercept is at the y = a.

Now to find the slope, take derivative of Eq. (*) with respect to x, on both sides.

(d/dx) lny = (d/dx) [lna + xlnb]

(1/y) dy/dx = lnb

or

dy/dx = y lnb

or

dy/dx = lnb ab^x, since y = ab^x.