Question

1. Why is e the most natural base for logarithms and exponential functions when it comes to calculus?

2. Outline how Carbon-14 dating is used to approximate the age of a fossil.

Answer 1

1.In mathematics, there is only one solution to the differential equation dx/dt = x with initial condition x=1.The solution is e^x, where e is a transcendental number.When base e is used in mathematics the answers come out naturally.Also because almost all natural phenomena posse an exponential law in their time evolution, almost every relevant differential equation has solutions of the form ekt , so ex is fundamental, thus its inverse function ln(x) too.When we model a practical problem whose rate of change a constant then this function comes as solution. That is why natural log is defined to the base to solve inverse problem.

2.The principle of carbon dating is simple: the rates at which various radioactive elements decay are known, and the ratio of the radioactive element to its decay products shows how long the radioactive element has existed in the rock. This rate is represented by the half-life, which is the time it takes for half of a sample to decay.The only way to measure the 14C in a sample was to detect the radioactive decay of individual carbon atoms.In this approach, what is measured is the activity, in number of decay events per unit mass per time period, of the sample. This method is also known as "beta counting", because it is the beta particles emitted by the decaying 14
C atoms that are detected.Half-life of Carbon-14: Radiometric dating is a technique used to date materials such as rocks or carbon, usually based on a comparison between the observed abundance of a naturally occurring radioactive isotope and its decay products, using known decay rates.
The half-life of carbon-14 is 5,730 years, so carbon dating is only relevant for dating fossils less than 60,000 years old. Radioactive elements are common only in rocks with a volcanic origin, so the only fossil-bearing rocks that can be dated radiometrically are volcanic ash layers. Carbon dating uses the decay of carbon-14 to estimate the age of organic materials, such as wood and leather.