Exponentially decaying variable I have a variable which decays very quickly (exponentially). I want to transform it in a way it maintains "exponentially decaying" property but which will allow me to control the "speed" of a decay.
original <- c(100, 80, 70, 65, 64, 63, 63, 63, 63, 62, 62, 61);
plot(original);
target <- c(100, 90, 70, 60, 65, 64, 64, 63, 63, 63, 63);

target is decaying much slower. I want to be able to controll for that.
How do I do that?
 A: Exponentiate it. 
If your variable decays exponentially then it decays like $y = e^{-\delta t}$, so the rate of decay is $\delta$. Well then $y^{\alpha} = e^{-\delta \alpha t}$ so the new rate of decay is $\delta \alpha$. Choose $\alpha$ to make that product as small or as large as you want. 
A: Rescale it.
Specifically, I am supposing that by "exponential decay" of $X$ you mean that when $x$ is sufficiently large, $\Pr(X \gt x) \sim \exp(-\lambda x)$ for a positive constant $\lambda$ which measures the "speed" of the decay.  Let $\alpha \gt 0$ and write $Y = \alpha X$ for the scaled version of $X$.  Then
$$\Pr(Y \gt y) = \Pr(\alpha X \gt y) = \Pr(X \gt y/\alpha) \sim \exp(-\lambda (y/\alpha)) = \exp(-(\lambda/\alpha) y),$$
showing that the speed of decay of $Y$ equals $\lambda/\alpha$.
Thus, if you want to transform $X$ to have a decay rate of $\mu$, solving for $\lambda/\alpha = \mu$ yields 
$$Y = \lambda X /\mu$$
as a solution.  (There are other transformations of $X$ having $\mu$ as a decay rate, but this is arguably the simplest.)
