Logarithmic mean

For distinct positive real numbers $a,b$, their logarithmic mean isusing 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, $L(1,4)=3/\log 4\approx 2.164$.

An integral representation explains its name and its relation to multiplicative interpolation: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 $c>0$.