How to apply a logarithmic trendline if X Values = 0?

Question

OK, I know its not possible because Log(0) is undefined. But here is my Problem:

I measured a Weight gain of my Test object's over Time. Test objects, made of nonwoven material, were placed in very humid Atmosphere (enclosed above a Tank of heated Water). As they absorbed kondensated Water their total Weight would increase. Later the increase in Weight was lesser than in the beginning since test objects became saturated.

I plotted a graph in excel with time as x axis and weight as y axis. All of my test objects followed a similiar pattern. In order to take measurements i had to remove them from this humid enviroment for a short time. But since water vaporized really quickly the outcome is distorted. One would expect a logarithmic weight gain, but in a short time objects started to absorb water so slowly that the weight gain was slower than the weight loss(vaporization). Therefore 3. and 4. Point should be slightly higher than they are now, had I not disturbed the Process with my measurements. This would result in a logarithmic weight gain.

I woudl like to explain it in my report and provide a Trendline, but since my first Measurment was made at t = 0 h excel cannot calculate it. Here is my question: What is the most scientifically correct way to generate such a Trendline? I see a few possible options:

1. Ignoring the first value (but i don't like that),
2. Adding 1 hour to whole experiment (Whatever happend at t = 0h would be presented as it happened at t = 1 h and so forth).

Or maybe there is another way? Any advice deeply appreciated.

P.S. I am writing my Thesis for Dipl. Ingenieur in Mechanical engineering in Germany.

EDIT: I added some additional info about my experiment.

EDIT2: I added pictures and added even some more info. I hope it's now clear enough;)

Answer 1

You've only got five points & a straight line looks like a good enough fit over their range. You say that points 3 & 4 should be higher owing to a defect of the measurement procedure but not why the other points shouldn't be higher by the same reasoning. So I'm not sure why you want to fit $\log w$ to $t$ (or to $\log t$), given that a model that predicts the test objects will get heavier without limit as time goes on isn't prima facie a very suitable one.

If you did want $\log w=\beta_0 + \beta_1 t +\varepsilon$, @user3170559's advice is fine. The model $\log w=\beta_0 + \beta_1 \log t +\varepsilon$ is problematic: as you've pointed out, $t$ takes values of zero. Sometimes people fudge this by adding a small constant to $t$, but the justification is that $t=0$ is impossible & really represents an imprecise measurement of a small value of $t$; not a justification you can appeal to in this case.

A better model would take into account prior engineering knowledge about the form relationships could take. $w=\beta_0 − \beta_1 \mathrm{e}^{−\beta_2 t} + \varepsilon$ would be the kind of thing I'd guess at—an engineer could probably do better—& the parameters might be meaningfully interpreted: $\beta_0$ as the saturated weight, $\beta_1$ as the range from dry to saturated, & $\beta_2$ as determining the rate of water uptake. You could reparameterize as appropriate for the things you're interested in—e.g. %age increase, initial weight, & rate—& compare the estimates for the test objects.

So it depends what you're going to do with the trend-lines. If you're not going to do anything you might as well just draw in by hand something that looks sensible.

[Throughout $w$ is weight, $t$ is time, $\beta$'s are the parameters (not the same ones in each model), & $\varepsilon$ the error.]

Answer 2

Why do you want to take the logarithm of the x-axis? If one observes data over time that seems to behave exponentially, one takes the logarithm of the observed data. So, I suggest you take the logarithm of the weights and plot this against time. Then you can just click on the plotted line of the graph, click on the line an add a trend line. After taking the logarithm you should take a linear trend line.