I have data that can be fitted using an exponentially decaying function: $y = e^{-t/\tau}$ and I want to determine the value of $\tau$.
I see that if I make the t-axis logarithmic I get a straight line, and my idea is to use this straight line to get the value of $\tau$.
First I tried to define a new variable $u = e^{t}$ so $t = ln(u)$ then I write
$y = e^{-ln(u) / \tau} = e^{-1/\tau} e^{ln(u)} = u e^{-1/\tau}$
so if I plot $y$ against $u$ I should get a straight line with gradient $e^{-1/\tau}$ and y-intercept $0$. But $u = e^t$ and I want to plot against $ln(t)$ not $e^t$, so this is wrong. Also, my gradient looked negative so the y-intercept should not be $0$.
So I tried to define the function the other way $u = ln(t)$ so $t = e^u$ but then my equation becomes
$y = e^{-e^u/\tau}$ which is not a straight line.
I'm very confused. Can anyone tell me what I'm doing wrong? Thank you!
You need to substitute: $z=\ln(y)$
$$y = e^{-t/\tau} \implies z=\ln(e^{-t/\tau}) \implies z = -\frac{1}{\tau}t$$
Plot $\ln(y)$ versus $t$ and the gradient will be: $-\dfrac{1}{\tau}$