Write a good notes about NATURAL LOGARITHMS AND ITS APPLICATIONS

Definition: If $e^y = x$, then $\ln(x) = y$.

Key Properties:

• $\ln(1) = 0$
• $\ln(e) = 1$
• $\ln(e^x) = x$
• $e^{\ln(x)} = x$ (for $x > 0$)
• $\ln(ab) = \ln(a) + \ln(b)$
• $\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)$
• $\ln(a^b) = b \ln(a)$

Applications:

• Exponential Growth and Decay: Used to model populations, radioactive decay, and compound interest (e.g., $A = P e^{rt}$).
• Calculus: The derivative of $\ln(x)$ is $\frac{1}{x}$, and the integral of $\frac{1}{x}$ is $\ln|x| + C$.
• Physics and Engineering: Appears in formulas for sound intensity (decibels), electrical circuits (RC circuits), and fluid dynamics.
• Statistics: Used in probability distributions, such as the log-normal distribution.