What is the application of natural logarithm in life?

Question

the application and the details such as equation (if included) and examples.

Answer 1

Engineers and Mathematicians use e^(x) and ln(x) quite often in working with different equations. One of the most common equations that I can think of is the equation used for exponential growth of a population. You could use it to figure out how many bacteria there will be x(t) in a certain amount of time (t) if you started out with x_0 amount of bacteria and they multiply at rate k

here is the function: x(t)=x_0 e^(kt)

sometimes it is written as Population Total = Population Initial * e^(rate * time) or Ptot= Pe^(rt)

Answer 2

In any situation where the rate of change is proportional to the magnitude of a variable, that magnitude will be an exponential function of that variable. For example: The rate of heat loss from an object is proportional to the temperature of that object. Then T = T0*e^-(t/tc), where T0 is the initial temperature and tc is the "time constant" that depends on the material properties of the object. The temperature is an exponential function of time. The time to reach a given temperature T involves a logarithm of the temperature. t = tc*ln(T0/T). There is basically no difference between natural and other logs, since one can be obtained from the other by a constant of multiplication. However, the mathematical derivations result in natural log functions.

Answer 3

The "Bell Curve," or Normal Probability Distribution is the basis for all statistics, and it is modelled based on Euler's number (~2.718, the base of Natural Logarithms). The width of the bell curve is determined by the standard deviation of the sample. The area under the bell curve = 1. The formula for the bell curve is:

y=1/(2*pi*Standard Dev.)^0.5 * e^ -(1/2)([x-Average]/Standard Dev.)^2

Answer 4

. . . and consider that the response of human sensory systems seems to be closely approximated by a logarithmic function. It appears we have it built in.