A natural logarithm is a logarithm with base \( e \), where \( e \approx 2.71828 \) is Euler's number. It is denoted by \( \ln N \), and is commonly used in calculus, mathematical modeling, and exponential growth or decay problems.
\[ \log_e N = \ln N \]
This expresses that the natural logarithm is the logarithm with base \( e \), and is written as \( \ln N \).
\[ \ln N = x \quad \text{implies} \quad e^x = N \]
This identity means that if the natural logarithm of \( N \) is \( x \), then \( e \) raised to the power of \( x \) equals \( N \).
\[ e = \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^n \approx 2.71828 \dots \]
This defines Euler’s number \( e \) as the limit of a sequence. It's an important constant in mathematics, especially in continuous growth models.