y = \ln x

Graph of the Natural Logarithm $y = \ln x$

$y = \ln x$ is the inverse of $e^x$ : it gives the power to which $e$ must be raised to obtain $x$ . Called the natural logarithm, it satisfies $e^{\ln x} = x$ .

Domain and range

Since $e^y$ only ever takes positive values, $\ln x$ makes sense only for $x > 0$ . The domain is $x > 0$ and the range is all real numbers: the domain and range of the exponential, exactly swapped.

Monotonicity and shape

The derivative is as follows.

y' = \frac{1}{x}

It is positive for $x > 0$ , so the function increases monotonically, but the slope approaches $0$ as $x$ grows, so the climb becomes ever gentler. The second derivative $y'' = -\dfrac{1}{x^2}$ is negative, so the curve is concave down throughout. It has no upper bound and rises forever, yet it does so remarkably slowly: $\ln 1000 \approx 6.9$ .

Asymptotes and limits

As $x \to 0^{+}$ the function diverges to $-\infty$ , so the $y$ -axis, the line $x = 0$ , is a vertical asymptote. As $x \to +\infty$ it diverges to $+\infty$ , but more slowly than any power of $x$ ; for every positive $p$ the following holds.

\lim_{x \to \infty} \frac{\ln x}{x^p} = 0

Notable points

The curve passes through $(1, 0)$ and $(e, 1) \approx (2.718, 1)$ . Since $a^0 = 1$ , every logarithm passes through $(1, 0)$ whatever its base. At $(1, 0)$ the tangent is the line $y = x - 1$ , of slope $1$ .

Laws of logarithms

Law	Formula	What it does
Product	$\ln(ab) = \ln a + \ln b$	turns multiplication into addition
Quotient	$\ln \dfrac{a}{b} = \ln a - \ln b$	turns division into subtraction
Power	$\ln a^p = p \ln a$	turns a power into multiplication

Lowering every operation by one level in this way is what made logarithms a computational tool for centuries.

Definition as an integral

The natural logarithm is also an antiderivative of $\dfrac{1}{x}$ .

\int_{1}^{x} \frac{dt}{t} = \ln x

This integral is sometimes taken as the definition of the natural logarithm. The change-of-base formula $\log_a x = \dfrac{\ln x}{\ln a}$ shows that every logarithm is a constant multiple of this one.

Applications

Statistics replaces a product of likelihoods with a sum of log-likelihoods
Information theory defines information content and entropy through logarithms
The $O(\log n)$ of computational complexity, as in binary search, measures algorithms that shrink the problem by a constant factor at every step

Graph of the Natural Logarithm y=ln⁡xy = \ln xy=lnx