Um, okay.
Anyone who has actually studied physics to the level of being familiar with the concepts of theoretical physics would know that exponential/logarithmic functions are quite simple and basic math which do not involve multivariate analysis. This is high school level math.
Also, anyone who is knowledgeable about theoretical physics knows that the rate of radioactive decay is invariable.
FYI, the reason I keep lumping exponents together with logarithms is that they are inverse functions--like multiplication is the inverse function of division.
Let me help you out a little: radioactive decay is described by the equation,
A=A0e-(kt/T1/2),
where A is the current quantity of the material,
A0 is the starting quantity of the material,
e is the "natural number", which has a value of ~2.71,
k is the decay constant which is ~0.693,
t is the elapsed time of decay (time since quantity A0 existed), and
T1/2 is the half-life of the material.
(The reason I put the tilde on those numbers is because they are truncated after 3 significant digits. They are like pi, in that they have an infinite number of digits after the decimal point.) In order to measure how old a sample is, all you need to do is to measure the quantities of the radioisotope and its decay products. From that, you can determine the starting quantity. (There are other ways to determine starting quantity, as well.) After that, it is just a matter of solving for t and plugging in numbers to determine how old the substance is. The technique is only inaccurate at the extremes: at the beginning, when too little time has elapsed to observe any radioactive decay, and at the end when there is too little radioisotope left to measure. In between those extremes, the technique is quite precise.
BTW, the mathematical formula used to calculate radioactive decay is the same formula used to calculate compound interest on your savings account. Have you ever seen the interest earned on your account jump around the way Answers In Genesis claims elapsed time determined by radioisotope dating jumps around? I'd go so far as to say that if the amount of interest your account earns is drastically different from month to month, then either the balance has significantly changed, or the interest rate has changed.
To express e, remember to memorize a sentence to simplify this...i.e. 2.7182818284
BTW, your argument begs the question.
When Rutherford used the Euler constant/Napier log to express a radioactive decay rate, he was simply beginning with a mathematical form associated with a decay rate that was NOT constant, but which was diminishing with time.
~0.693 is simply the natural log of 2, i.e. for a half life of that mathematical form used to express decay. k, t, and tau might be very complicated expressions for any given material and situation.
The form he used doesn’t address a beginning point, but instead assumes the function is infinite in both directions. He probably should have begun with Laplace transforms since they begin at t=0.
The formula used as a radioactive decay rate, assumes a material will statistically decay at a rate proportional to the remaining unstable states within that material.
I don’t think I would hinge my eternal destiny based upon such an assumption.
The use of natural logs is simply a mathematical tool used to simplify equations in the identification problem. They are only as useful as they might identify with actual measurables. It’s important to understand the assumptions made in those expressions, what they mean, and how they are identifiable or translatable to physical phenomenon.
Ask an accountant about how many different ways one can compound interest. The same might be said about radioactive decay, and it isn’t always simply calculated.
A more fruitful study is probably to study Scripture to see what God provides us, indicating when a Dirac delta or Heaviside jump isn’t more appropriate in the initial problem formulation of historical physical laws.