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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 $c>0$.

Both injective and surjective. The natural logarithm is strictly increasing on the positive real numbers, so it is injective. For every $y\in \mathbb{R}$, $x=e^y>0$ satisfies $\ln x=y$; hence the function is surjective, with the exponential function as its inverse function.

A positive real number is a real number $x$ with $x>0$. Zero is excluded; allowing zero gives a nonnegative real number. Both $1/2$ and $\sqrt{2}$ are positive.

The sum, product, and reciprocal of positive real numbers are positive. Consequently the positive real numbers form a group under multiplication, and the natural logarithm converts that multiplication into addition: $\log (xy)=\log x+\log y$. Positivity is also the domain restriction that makes the real logarithm well-defined.

For a nonnegative first-passage time $T_A$, possibly infinite, the survival probability isIt describes the Probability that absorption has not yet occurred. The tail-integral identityholds as an identity of extended nonnegative quantities, even when absorption is not almost sure or the expected time is infinite. Indeed, $T_A={\int }_0^{\infty }{1}_{\{T_A>t\}}dt$ pathwise, and Tonelli's theorem exchanges expectation and integration. For an integer-valued hitting time, the corresponding formula is ${\mathbb{E}}_x{T}_A={\sum }_{n=0}^{\infty }{\mathbb{P}}_x(T_A>n)$.

If the finite-time distribution is absolutely continuous, its first-passage-time density is $-{\partial }_{t}Q(x,t)$. Where $Q>0$, its hazard function is $-{\partial }_t\log Q$. Constant hazard $\lambda >0$ gives $Q(t)=e^{-\lambda t}$ and mean $1/\lambda$.

The limit of $Q(x,t)$ as $t\to \infty$ is the Probability of never being absorbed; a positive limit forces infinite mean. Conversely, zero limit does not guarantee a finite mean. Brownian first passage to a positive level occurs almost surely, but its survival tail is proportional to $t^{-1/2}$, whose integral diverges.

The up game is the short partizan game $\uparrow =\{0\mid \ast \}$, where $\ast =\{0\mid 0\}$ is the star game. If Left starts, moving to zero wins. If Right starts, the only move leaves star to Left, who again moves to zero and wins. Thus up is strictly positive in the game ordering.

Its value is nevertheless smaller than every positive real number: it is an infinitesimal combinatorial game. This makes it useful for separating numerical advantage from a tiny advantage that can decide an otherwise balanced disjunctive sum. Reversing the players gives down, $\downarrow =\{\ast \mid 0\}=-\uparrow$.