why is the geometric mean less than the logarithmic mean? Can someone explain why the geometric mean is less than the logarithmic mean?
$$\sqrt{ab} \leq  \frac{b-a}{\log b-\log a}
$$

This not an answer, but this remark cannot take place in a simple comment.
It should be said that, under the form $\dfrac{a-b}{ln(a)-ln(b)}$, the logarithmic mean may look rather artificial and difficult to extend to more than 2 variables unless one knows the formula 
$$L(a,b)=\int_0^1 \ \varphi(x)dx \ \ \text{with} \ \ \varphi(x)=a^xb^{1-x}=be^{x \ ln(a/b)}$$
which extends to 3D under the form:
$$L(a,b,c)=2\int_{x=0}^1\int_{y=0}^{1-x} a^x \ b^y \ c^{1-x-y} dx dy$$
and more generally in nD in the same way on the simplex defined by $x_1+x_2+\cdots +x_n=1$ (factor $2=2!$ will become $(n-1)!$). 
So, the next question is how one can prove for example that 
$$G(a,b,c)\leq L(a,b,c) \ ?$$
