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For a positive random variable $X$, let $D\in \{1,\dots ,9\}$ be its first nonzero base-ten digit. The Benford law is the discrete probability distribution with probability mass functionHere $X$ need not be an integer: $0.0042$, $4.2$ and $4200$ all have $D=4$. Multiplication by an integer power of ten changes the decimal point's position and leaves the first nonzero digit unchanged. Zero has no first nonzero digit, so exclude it. For negative inputs one may explicitly use their absolute values, still excluding zero; that convention must be stated rather than applying a logarithm to a negative number.

The Benford law favors small digits: $\mathbb{P}(D=1)={\log }_{10}2\approx 0.301029996$, whereas $\mathbb{P}(D=9)={\log }_{10}(10/9)\approx 0.04575749$. Thus the model assigns about $30.10\%$ to digit one and $4.58\%$ to digit nine. The masses are positive and their normalization is a finite telescoping series:

Why can digit one be so common? In a decimal significand interval $[1,10)$, the digit-one region $[1,2)$ occupies a larger fraction of the logarithmic scale than the digit-nine region $[9,10)$. The precise assumption is that the fractional part of the base-ten logarithm is distributed uniformly on $[0,1)$; Benford law from logarithmic uniformity turns those unequal interval lengths into probabilities. This makes a surprising digit pattern into a simple geometry of multiplicative size. The logarithmic explanation is also presented in Peter Sheridan Dodds, Benford's Law, essential story.

The Benford law is useful as a proposed model for leading digits when the mechanism supports logarithmic uniformity or an appropriate limiting argument, such as Benford frequencies for powers of an integer. Merely spanning many powers of ten does not establish that assumption. A discrepancy from the Benford law can prompt investigation of the model and data collection; by itself it does not establish fraud, and agreement does not establish authenticity.

Simon Newcomb's 1881 paper, Note on the Frequency of Use of the Different Digits in Natural Numbers, appeared in the American Journal of Mathematics 4, pp. 39–40. Its opening connects unequal leading-digit frequencies with the faster wear of early pages in logarithm tables. The publication metadata and original opening excerpt are available in Newcomb (1881), original publication record and ^[ref]. Frank Benford's The Law of Anomalous Numbers, Proceedings of the American Philosophical Society 78 (1938), pp. 551–572, likewise discusses worn logarithm tables on p. 551 and tabulates observations on p. 553. See Benford (1938), transcription of the original paper with page numbers and publication metadata. These observations motivated study of the digit pattern; they do not make the Benford law universal.

A continuous transformation $T$ of a compact metric space $X$ is uniquely ergodic when there is exactly one $T$-invariant Borel probability measure on $X$. We use the usual compact-space convention for unique ergodicity; the compactness and continuity hypotheses matter in the assertion about all starting points.

Let $R_{\alpha }(x)=x+\alpha (\mathrm{mod}1)$ on the circle group, with $\alpha$ irrational. Normalized Lebesgue measure $m$ is invariant under this irrational rotation of the circle. If $\nu$ is any invariant Borel probability measure, define its Fourier coefficients by $c_k=\int e^{2\pi ikx}d\nu (x)$. Invariance givesFor $k\ne 0$, irrationality makes $e^{2\pi ik\alpha }\ne 1$, so $c_k=0$; also $c_0=1$. These are the Fourier coefficients of $m$. Hence $\nu$ and $m$ have the same integrals against every trigonometric polynomial. Such polynomials are uniformly dense in the continuous functions by the Stone-Weierstrass theorem, so the measures agree on every continuous test function and therefore agree as Borel measures. Thus

For the general uniquely ergodic system, fix $x\in X$ and form the empirical measuresOn a compact metric space, the Borel probability measures are compact for weak convergence of probability measures. Any subsequential limit $\nu$ is invariant: for every continuous $g$,Here $g$ is bounded and $g\circ T$ is continuous, so the identity passes to the limit. Uniqueness of the invariant Borel probability measure gives $\nu =\mu$. Every subsequential limit is therefore $\mu$, and compactness implies convergence of the entire sequence. Testing against $f$ provesIn fact this proves uniform ergodic convergence for uniquely ergodic systems: if convergence were not uniform in $x$, choose $N_j\to \infty$ and $x_j$ where the discrepancy stays above a fixed positive number. The same compactness and telescoping argument applied to ${\nu }_{N_j,x_j}$ forces a subsequence to converge to $\mu$, a contradiction. This also makes clear why an almost-everywhere Birkhoff ergodic theorem alone would not establish the requested everywhere assertion. Without the compact-space hypothesis the assertion need not hold: on the discrete space $\{p\}\bigsqcup {\mathbb{Z}}_{\ge 0}$, set $T(p)=p$ and $T(n)=n+1$. The only invariant probability is ${\delta }_p$, but the continuous bounded function which is zero at $p$ and one on the integer orbit has orbit average one there.

For the decimal application put $\alpha ={\log }_{10}2$. This is irrational: if $\alpha =p/q$ with positive integers $p,q$, then $2^q=10^p=2^p{5}^p$, contradicting unique prime factorization. Writing $n\alpha =m_n+t_n$ with $m_n=\lfloor n\alpha \rfloor$ and $t_n\in [0,1)$ gives $2^n=10^{m_n}10^{t_n}$. Its leading decimal digit is seven exactly whenThe half-open upper endpoint correctly excludes powers whose leading digit is eight, including $2^3=8$.

The indicator of $I$ is not continuous, so an extra step is needed. For any $\epsilon >0$, choose continuous functions $a_{\epsilon },b_{\epsilon }$ on the circle with $0\le a_{\epsilon }\le 1_I\le b_{\epsilon }\le 1$ and $\int (b_{\epsilon }-a_{\epsilon })dm<\epsilon$, by tapering in small neighbourhoods of the two endpoints. Applying the everywhere averaging result to these functions traps the lower and upper limits of the interval frequency between their integrals. Letting $\epsilon \downarrow 0$ proves everywhere interval frequency under an irrational rotation. ConsequentlyThis is the leading-seven case of Benford frequencies for powers of an integer.