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For a positive random variable $X$, write its base-ten logarithm as an integer part plus a fractional part:The floor function handles values below one as well: for $X=0.0042$, $K=-3$ and the decimal significand is $10^Z=4.2$. Since $10^Z\in [1,10)$, the first nonzero decimal digit is $D=\lfloor 10^Z\rfloor$. The monotonicity of the logarithm gives the exact event equivalenceIf $Z$ has the uniform distribution on $[0,1)$, the probability of this interval is its length. Consequently $\mathbb{P}(D=d)={\log }_{10}(d+1)-{\log }_{10}d$, proving the Benford law. This argument assumes uniformity of the fractional part of the logarithm, rather than uniformity of $X$ itself. See the logarithmic-coordinate explanation in Peter Sheridan Dodds, Benford's Law, essential story.

An exact constructed example is the log-uniform distribution on $[1,10)$, with probability density function $f_X(x)=1/(x\ln 10)$ there and zero elsewhere. Its total mass is ${\int }_1^{10}dx/(x\ln 10)=1$. The change of variables formula with $x=10^z$ gives $f_Z(z)=f_X(10^z)(\ln 10)10^z=1$ for $0\le z<1$. Equivalently, integrating directly givesFor example, the probability of first digit four is ${\log }_{10}(5/4)\approx 0.096910013$. By contrast, an ordinary uniform distribution for $X$ on $[1,10)$ has probability density function $1/9$ there, so $\mathbb{P}(D=d)={\int }_d^{d+1}dx/9=1/9$ for every digit. This is a concrete counterexample to any claim that the Benford law holds for every positive random variable.

More generally, a log-uniform distribution on $[A,B)$ with $0<A<B$ has uniform $Y={\log }_{10}X$ on $[{\log }_{10}A,{\log }_{10}B)$. If its logarithmic span ${\log }_{10}(B/A)=m$ is a positive integer, reduction to the fractional part wraps an interval of length $m$ around the unit interval exactly $m$ times, making $Z$ uniform even when $A$ is not a power of ten. Arbitrary partial-decade truncations generally do not have this property: for a log-uniform distribution on $[1,2)$, the first digit is always one.

Under the uniform fractional-logarithm assumption, multiplication by any constant $c>0$ replaces $Z$ by $Z^{\prime }=\{Z+{\log }_{10}c\}$. This is a rotation modulo one, which preserves the uniform distribution and therefore preserves the Benford law; see scale invariance of decimal significands. For the constructed $[1,10)$ example, multiplying by $0.001$ gives a log-uniform distribution on $[0.001,0.01)$ with the same digit probabilities. The claim concerns the decimal significand law: there is no proper probability density function proportional to $1/x$ over all of $(0,\infty )$, since its integral diverges. The existing scale-invariant prior describes that unnormalizable measure, while a finite log-uniform distribution is normalizable.

Changing currency or monetary units multiplies amounts by a common constant without changing their generating mechanism. This motivates scale invariance of decimal significands of a generic significand distribution.

Multiplication by $c$ adds ${\log }_{10}c$ to the logarithm, rotating its fractional part. A probability distribution on the unit circle invariant under every rotation must be uniform: equal-length arcs have equal mass, and subdivision fixes that mass to arc length. Uniform fractional logarithms then give Benford law. Thus unit or currency invariance motivates uniform logarithmic mantissas. Restricted ranges, prescribed thresholds and rounding can prevent a particular collection from having this invariance.