Natural Logarithm
Functions & Advanced AlgebraThe 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 just a logarithm with a special base: the number $e \approx 2.718$. It answers: "What power of $e$ gives me $x$?"
Example
$\ln(e) = 1$ because $e^1 = e$. $\ln(e^3) = 3$. $\ln(1) = 0$ because $e^0 = 1$. On a calculator, the "ln" button computes this.
Key Insight
The number $e$ is the base that makes calculus simplest. The natural logarithm shows up constantly in science because the derivative of $\ln(x)$ is simply $1/x$, the cleanest possible result.
Definition
The natural logarithm $\ln(x) = \log_e(x)$ is the inverse of $e^x$. Domain: $x > 0$. Range: all reals. Properties: $\ln(xy) = \ln(x) + \ln(y)$, $\ln(x/y) = \ln(x) - \ln(y)$, $\ln(x^r) = r\ln(x)$, $\ln(e) = 1$, $\ln(1) = 0$.
Example
Solve $e^{2x} = 10$: take ln of both sides: $2x = \ln(10) \approx 2.303$. So $x \approx 1.151$. Find doubling time for $8\%$ continuous growth: $T_d = \ln(2)/0.08 \approx 8.66$ years.
Key Insight
Natural logarithms appear in formulas for continuous compounding, exponential growth/decay, entropy in information theory, and probability distributions (e.g., the lognormal distribution). When in doubt in advanced math, use ln.
Definition
The natural logarithm is defined analytically by $\ln(x) = \int_1^x (1/t)\,dt$ for $x > 0$. It satisfies $d/dx[\ln(x)] = 1/x$ and is the unique antiderivative of $1/x$ mapping $1$ to $0$. The complex extension: $\text{Ln}(z) = \ln|z| + i\,\text{Arg}(z)$ for $z \in \mathbb{C} \setminus \{\text{non-positive reals}\}$.
Example
Sterling's approximation: $n! \approx \sqrt{2\pi n} \cdot (n/e)^n$. Taking the natural log: $\ln(n!) \approx n\ln(n) - n + (1/2)\ln(2\pi n)$. This is derived using $\ln(n!) = \sum \ln(k) \approx \int_1^n \ln(t)\,dt$.
Key Insight
The natural logarithm is "natural" because it is the unique logarithm whose derivative at $1$ equals $1$. All other logarithms differ by a constant factor (the scaling $\ln(b)$). This uniqueness makes it the canonical choice in analysis.