Mathematics · Calculus

Natural Logarithm

Reference entry · last updated October 12, 2026

The natural logarithm, written \(\ln x\), is the logarithm to base Euler’s number \(e\). For a positive real number \(x\), \(\ln x\) is the real exponent to which \(e\) must be raised to obtain \(x\).[1, 2]

First principles and definitions

The inverse relation between the natural logarithm and the natural exponential is:[2]

\[y=\ln x\ \Longleftrightarrow\ e^y=x,\qquad x>0,\ y\in\mathbb{R}.\]

An equivalent definition uses an integral:[1, 3]

\[\ln x=\int_1^x\frac{dt}{t},\qquad x>0.\]

For \(x>1\), this is the area under \(1/t\) from \(1\) to \(x\). For \(0<x<1\), it is the negative of the area from \(x\) to \(1\). Thus \(\ln1=0\), while \(\ln e=1\).[1, 3]

Domain, range, and graph

The real domain is \((0,\infty)\), and the range is all real numbers. The graph passes through \((1,0)\) and \((e,1)\). It increases throughout its domain.[2, 3]

Its slope \(1/x\) decreases as \(x\) increases, so the graph is concave down. The line \(x=0\) is a vertical asymptote:[3]

\[\lim_{x\to0^+}\ln x=-\infty,\qquad \lim_{x\to\infty}\ln x=\infty.\]

The graph of \(\ln x\) for positive \(x\). Values below \(1\) have negative logarithms.

Logarithm laws

For positive real \(a,b\) and real \(r\), the logarithm laws are:[2]

Operation	Identity
Product	\(\ln(ab)=\ln a+\ln b\)
Quotient	\(\ln(a/b)=\ln a-\ln b\)
Power	\(\ln(a^r)=r\ln a\)

The inverse identities are \(\ln(e^x)=x\) for real \(x\) and \(e^{\ln x}=x\) for \(x>0\).[2]

Derivatives, integrals, and change of base

The fundamental theorem of calculus gives the derivative. The chain rule extends it to a differentiable function \(g\):[3, 4]

\[\frac{d}{dx}\ln x=\frac1x\quad(x>0),\qquad \frac{d}{dx}\ln g(x)=\frac{g'(x)}{g(x)}\quad(g(x)>0).\]

An antiderivative of \(1/x\) uses absolute value. The formula holds on any interval that does not cross zero:[3]

\[\int\frac{dx}{x}=\ln|x|+C,\qquad x\ne0.\]

Integration by parts with \(u=\ln x\) and \(dv=dx\) gives:[5]

\[\int\ln x\,dx=x\ln x-x+C,\qquad x>0.\]

For \(x>0\) and a real base \(b>0\), \(b\ne1\), change of base gives:[1, 2]

\[\log_b x=\frac{\ln x}{\ln b}.\]

Algebra example and common errors

Illustrative example: solving \(e^{2x}=5\) by taking natural logarithms gives \(2x=\ln5\), so \(x=\tfrac12\ln5\).

The real logarithm of zero or a negative number is undefined. A limit tending to \(-\infty\) does not assign a value to \(\ln0\).[2, 3]
Logarithms do not distribute over addition. For example, \(\ln(1+1)=\ln2\), while \(\ln1+\ln1=0\).[2]
For real \(x\ne0\), \(\ln(x^2)=2\ln|x|\). Writing \(2\ln x\) restricts the expression to \(x>0\).[2, 3]

References

NIST, Digital Library of Mathematical Functions, §4.2: Definitions, equations 4.2.8 and 4.2.11–14.
OpenStax, Calculus Volume 1, §1.5: Exponential and Logarithmic Functions, logarithmic functions and change-of-base formulas.
OpenStax, Calculus Volume 1, §6.7: Integrals, Exponential Functions, and Logarithms, definition 6.24 and theorems 6.15–18.
OpenStax, Calculus Volume 1, §3.9: Derivatives of Exponential and Logarithmic Functions, theorem 3.15.
OpenStax, Calculus Volume 2, §3.1: Integration by Parts, theorem 3.1.