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.
Absolute non-sequitur.
Reality says one thing, even down to the fundamental constants of the universe. The Bible says another. My eternal destiny does not depend on my ability to disbelieve reality in favor of literally believing the Bible. Furthermore, since the physical, observable universe is what God made, to believe that it is an illusion and only the description given in the Bible is accurate is to believe that God is a liar. I do not consider a lying God worthy of worship. The Bible is a moral guide, and the truths it tells are moral--not physical--truths.
FYI, just because a simple equation has a rather complicated derivation does not mean that it is necessary for the derivation to be performed every time one wants to calculate a value with that equation. The only reason students are walked through such derivations is so that they understand the basis of the equation.
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.
Let's be absolutely clear here: radioactive decay is not linear, but it *is* constant. A graph of a quantity that decreases by half at a specific time interval does not form a straight line, but a logarithmic curve.
To express e, remember to memorize a sentence to simplify this...i.e. 2.7182818284
That's rather nonsensical, and it is also beside the point. To figure out how old a sample is, it is not necessary to know e out to 50 digits... it is only necessary to accurately measure the amount of radioactivity in the sample. In fact, if you are solving for t (the age of the sample) in the equation I previously posted, you do not use e at all, since solving for t requires converting e to its logarithmic function.
Also, I should point out that extending e out to any number of digits does not significantly alter the length of time that a radioisotope has been decaying. You earlier claimed that there were huge errors in measurements using radiometric dating techniques--whether you use e to three significant digits or three hundred, it has little effect on the calculation. A determination of 1.5 million years is not significantly different than 1.49285693 million years.
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.
We live in a world of logarithms; our brains are hard-wired to perceive the world through a logarithmic function. That means that logarithms are not "mathematical tools", but are the basic language of our existence.