= Logarithmic mean
{title2=$L(a,b)$}
For distinct <positive real numbers> $a,b$, their logarithmic mean is
$$
L(a,b)=\frac{b-a}{\log b-\log a},
$$
using the <natural logarithm>. Set $L(a,a)=a$ by continuity: the difference quotient of the <natural logarithm> tends to its <derivative> $1/a$, so the reciprocal tends to $a$. For example, $L(1,4)=3/\log4\approx2.164$.
An <integral> representation explains its name and its relation to multiplicative interpolation:
$$
L(a,b)=\int_0^1 a^{1-t}b^t\,dt.
$$
For $a\ne b$, integrate $a\exp(t\log(b/a))$ directly; for $a=b$, the integrand is constant. Every interpolated value lies between $a$ and $b$, so the logarithmic mean does too. It is symmetric in $a,b$ and homogeneous: $L(ca,cb)=cL(a,b)$ for $c>0$.
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