Percent-wise decay in exponential function

Question

I've been given a set of points, and I used Geogebra's FitExp tool to perform an exponential regression to find a suitable model for the decay in the values.

I ended up with the function $f(x) = 40308e^{-0.05x}$, which visually seems to fit pretty well.

$x$ is counted in years, which will become important in a moment.

Question

Given this function, what is the percentage decay per year?

My thoughts

I'm thinking, given $-0.05x$ in the exponent, that it decays by 5% each year, but I can't make a solid argument for it.

I've found $\frac{f(1)}{f(0)} = 0.95 = 1-0.05$ if that helps, but still, how do I firmly state AND reason why -5% is the rate of decay for this function?

Thanks in advance for any help!

Answer 1

The relative change is given by $$\frac{f(x+1)-f(x)}{f(x)} = e^{-0.05}-1 \approx -0.0487705755.$$

Hence, the function decays by approximately $4.87705755\%$ every year.

Answer 2

Another way to think about it as by trying to conceptualize this in terms of a standard financial growth formula vs the general exponential form you have (I am signifying the difference between the r terms because they are not the same number):

$$y = C(r_f-1)^t$$

$$y = Ce^{rt}$$

These are identical. We know from exponent identities that:

$$(a^b)^c = a^{bc}$$

Therefore we can rewrite your general form as this:

$$y = C(e^r)^t$$

You can see now that

$$ e^r = 1 - r_f $$ $$ r_f = 1 - e^r$$