Natural Logarithm
The natural logarithm, written ln(x), is the logarithm with base e (approximately 2.718) and is the inverse of the natural exponential function.
Formula
\ln(x) = \log_e(x)
Definition
The natural logarithm, written $\ln$, is a logarithm with the special base $e \approx 2.718$: $\ln(x) = \log_e(x)$ answers "what power of $e$ gives $x$?", is the inverse of $e^x$, with domain $x > 0$ and range all reals. It satisfies $\ln(xy) = \ln(x) + \ln(y)$, $\ln(x/y) = \ln(x) - \ln(y)$, $\ln(x^r) = r\ln(x)$, $\ln(e) = 1$, and $\ln(1) = 0$. Analytically, $\ln(x) = \int_1^x (1/t)\,dt$ for $x > 0$, satisfying $d/dx[\ln(x)] = 1/x$ as the unique antiderivative of $1/x$ mapping $1$ to $0$; its complex extension is $\text{Ln}(z) = \ln|z| + i\,\text{Arg}(z)$ for $z \in \mathbb{C}$ excluding non-positive reals.
Example
$\ln(e) = 1$ because $e^1 = e$, $\ln(e^3) = 3$, and $\ln(1) = 0$ because $e^0 = 1$. To solve $e^{2x} = 10$, take $\ln$ of both sides: $2x = \ln(10) \approx 2.303$, so $x \approx 1.151$; finding the doubling time for $8\%$ continuous growth gives $T_d = \ln(2)/0.08 \approx 8.66$ years. Sterling's approximation, $n! \approx \sqrt{2\pi n} \cdot (n/e)^n$, taking the natural log becomes $\ln(n!) \approx n\ln(n) - n + (1/2)\ln(2\pi n)$, derived using $\ln(n!) = \sum \ln(k) \approx \int_1^n \ln(t)\,dt$.
Key Insight
The number $e$ is the base that makes calculus simplest, and the natural logarithm shows up constantly in science because the derivative of $\ln(x)$ is simply $1/x$, the cleanest possible result; it appears in formulas for continuous compounding, exponential growth and decay, entropy in information theory, and probability distributions like the lognormal. The natural logarithm is "natural" because it is the unique logarithm whose derivative at $1$ equals $1$: all other logarithms differ from it by a constant scaling factor, $\ln(b)$, which is what makes it the canonical choice in analysis.