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This is a pretty vague question, but I take it that you're groping for some "physical significance". The clearest one is that the logarithm is the inverse of the exponential function $x\mapsto e^x$ which itself arises whenever the rate of quantity's variation is equal to or proportional to that quantity, a fairly common statement describing physical processes: four example rates of chemical reactions, radio active decays, attenuation of light or other EM radiation through mediums. Given this "physical definition" it follows then that the inverse function is simply that given by $x\mapsto \int_1^x \frac{\mathrm{d}z}{z}$ and then this definition is broadened into the punctured complex plane $\mathbb{C}\sim \{0\}$ by analytic continuation. Moreover the functions $\exp$ and $\log$ defined in this way have particularly simple Taylor series (the former is universally convergent, the latter convergent in an open unit radius circle about $z=1$) that make their definitions relatively easy to broaden to object other numbers such as matrices, operators and so forth.

The idea of a rate of a quantity's variation being proportional to that quantity is further generalized in operator equations and, in particular, in the theory of Lie groups, where $\exp$ and its inverse $\log$ play central roles in mapping neighbourhoods of the group's identity to and from the "Lie algebra", i.e. the space of the linear transformations that play the role of generalized "rates of change" - these can now be complex numbers, quaternions or in general square matrices (for the Lie algebra they can always be thought of as square matrices - Ado's theorem - but this is not always so for the Lie group). Again, it is the natural base $e$ logarithm that falls from the definitions by dint of its Taylor expansion around the identity. The theory of Lie groups, with its fundamental reliance on $\exp$ and $\log$, plays many important roles in physics and the sciences in general. In an even more generalized setting, the Schrödinger equation is also a generalized "rate of change proportional to the quantity" equation, as are the descriptions of flows and the exponential map defining geodesics in differential geometry.

Lastly, since you ask about thermodynamics and the formula graven on Boltzmann's headstone, the logarithm is the grounding of the natural encoding of the idea that numbers of possibilities (volumes of phase spaces) multiply, whereas intuitively the corresponding "entropies" should add. Whilst it should be clear that the logarithm's base does not matter for this definition (indeed information theorists choose base 2 logarithms to write informational entropies in bi nary digi ts or bits), one could argue that the natural base $e$ logarithm that is the "prototypical" isomorphism between the group of reals and addition and the group of strictly positive reals and multiplication (which is what Boltzmann's intuitive idea is all about) that arises from the Lie theoretical idea of mapping the Lie group $(\mathbb{R}^+\sim\{0\},\,\times)$ onto its Lie algebra $(\mathbb{R},\,+)$

What is the probability of all this happening? It's precisely equal to unity: for the above ideas are how we define the natural logarithm (i.e. as the ones defined above as opposed to logarithms with another base or even indeed another function altogether).

This is a pretty vague question, but I take it that you're groping for some "physical significance". The clearest one is that the logarithm is the inverse of the exponential function $x\mapsto e^x$ which itself arises whenever the rate of quantity's variation is equal to or proportional to that quantity, a fairly common statement describing physical processes. For example: rates of chemical reactions, radio active decays, attenuation of light or other EM radiation through mediums all follow such laws. Given this "physical definition" it follows then that the inverse function is simply that given by $x\mapsto \int_1^x \frac{\mathrm{d}z}{z}$ and then this definition is broadened into the punctured complex plane $\mathbb{C}\sim \{0\}$ by analytic continuation. Moreover the functions $\exp$ and $\log$ defined in this way have particularly simple Taylor series (the former is universally convergent, the latter convergent in an open unit radius circle about $z=1$) that make their definitions relatively easy to broaden to objects other than numbers such as matrices, operators and so forth.

The idea of a rate of a quantity's variation being proportional to that quantity is further generalized in operator equations and, in particular, in the theory of Lie groups, where $\exp$ and its inverse $\log$ play central roles in mapping neighbourhoods of the group's identity to and from the "Lie algebra", i.e. the space of the linear transformations that play the role of generalized "rates of change" - these can now be complex numbers, quaternions or in general square matrices (for the Lie algebra they can always be thought of as square matrices - Ado's theorem - but this is not always so for the Lie group). Again, it is the natural base $e$ logarithm that falls from the definitions by dint of its Taylor expansion around the identity. The theory of Lie groups, with its fundamental reliance on $\exp$ and $\log$, plays many important roles in physics and the sciences in general. In an even more generalized setting, the Schrödinger equation is also a generalized "rate of change proportional to the quantity" equation, as are the descriptions of flows and the exponential map defining geodesics in differential geometry.

Lastly, since you ask about thermodynamics and the formula graven on Boltzmann's headstone, the logarithm is the grounding of the natural encoding of the idea that numbers of possibilities (volumes of phase spaces) multiply, whereas intuitively the corresponding "entropies", as extensive protperties of thermodynamic systems should add. Whilst it should be clear that the logarithm's base does not matter for this definition (indeed information theorists choose base 2 logarithms to write informational entropies in bi nary digi ts or bits), one could argue that the natural base $e$ logarithm that is the "prototypical" isomorphism (which is what Boltzmann's intuitive idea is all about) between the group of reals and addition and the group of strictly positive reals and multiplication that arises from the Lie theoretical idea of mapping the Lie group $(\mathbb{R}^+\sim\{0\},\,\times)$ onto its Lie algebra $(\mathbb{R},\,+)$

What is the probability of all this happening? It's precisely equal to unity: for the above ideas are how we define the natural logarithm (i.e. as the ones defined above as opposed to logarithms with another base or even indeed other functions altogether).

This is a pretty vague question, but I take it that you're groping for some "physical significance". The clearest one is that the logarithm is the inverse of the exponential function $x\mapsto e^x$ which itself arises whenever the rate of quantities variation is equal to or proportional to that quantity, a fairly common statement describing physical processes: four example rates of chemical reactions, radio active decays, attenuation of light or other EM radiation through mediums. Given this "physical definition" it follows then that the inverse function is simply that given by $x\mapsto \int_1^x \frac{\mathrm{d}z}{z}$ and then this definition is broadened into the punctured complex plane $\mathbb{C}\sim \{0\}$ by analytic continuation. Moreover the functions $\exp$ and $\log$ defined in this way have particularly simple Taylor series (the former is universally convergent, the latter convergent in an open unit radius circle about $z=1$) that make their definitions relatively easy to broaden to object other numbers such as matrices, operators and so forth.

What is the probability of this happening? It's precisely equal to unity: for that is how we define the natural logarithm (i.e. as the one defined above as opposed to a logarithm with another base or even indeed another function altogether).

This is a pretty vague question, but I take it that you're groping for some "physical significance". The clearest one is that the logarithm is the inverse of the exponential function $x\mapsto e^x$ which itself arises whenever the rate of quantity's variation is equal to or proportional to that quantity, a fairly common statement describing physical processes: four example rates of chemical reactions, radio active decays, attenuation of light or other EM radiation through mediums. Given this "physical definition" it follows then that the inverse function is simply that given by $x\mapsto \int_1^x \frac{\mathrm{d}z}{z}$ and then this definition is broadened into the punctured complex plane $\mathbb{C}\sim \{0\}$ by analytic continuation. Moreover the functions $\exp$ and $\log$ defined in this way have particularly simple Taylor series (the former is universally convergent, the latter convergent in an open unit radius circle about $z=1$) that make their definitions relatively easy to broaden to object other numbers such as matrices, operators and so forth.

The idea of a rate of a quantity's variation being proportional to that quantity is further generalized in operator equations and, in particular, in the theory of Lie groups, where $\exp$ and its inverse $\log$ play central roles in mapping neighbourhoods of the group's identity to and from the "Lie algebra", i.e. the space of the linear transformations that play the role of generalized "rates of change" - these can now be complex numbers, quaternions or in general square matrices (for the Lie algebra they can always be thought of as square matrices - Ado's theorem - but this is not always so for the Lie group). Again, it is the natural base $e$ logarithm that falls from the definitions by dint of its Taylor expansion around the identity. The theory of Lie groups, with its fundamental reliance on $\exp$ and $\log$, plays many important roles in physics and the sciences in general. In an even more generalized setting, the Schrödinger equation is also a generalized "rate of change proportional to the quantity" equation, as are the descriptions of flows and the exponential map defining geodesics in differential geometry.

Lastly, since you ask about thermodynamics and the formula graven on Boltzmann's headstone, the logarithm is the grounding of the natural encoding of the idea that numbers of possibilities (volumes of phase spaces) multiply, whereas intuitively the corresponding "entropies" should add. Whilst it should be clear that the logarithm's base does not matter for this definition (indeed information theorists choose base 2 logarithms to write informational entropies in bi nary digi ts or bits), one could argue that the natural base $e$ logarithm that is the "prototypical" isomorphism between the group of reals and addition and the group of strictly positive reals and multiplication (which is what Boltzmann's intuitive idea is all about) that arises from the Lie theoretical idea of mapping the Lie group $(\mathbb{R}^+\sim\{0\},\,\times)$ onto its Lie algebra $(\mathbb{R},\,+)$

What is the probability of all this happening? It's precisely equal to unity: for the above ideas are how we define the natural logarithm (i.e. as the ones defined above as opposed to logarithms with another base or even indeed another function altogether).

This is a pretty vague question, but I take it that you're groping for some "physical significance". The clearest one is that the logarithm is the inverse of the exponential function $x\mapsto e^x$ which itself arises whenever the rate of quantities variation is equal to or proportional to that quantity, a fairly common statement describing physical processes: four example rates of chemical reactions, radio active decays, attenuation of light or other EM radiation through mediums. Given this "physical definition" it follows then that the inverse function is simply that given by $x\mapsto \int_1^x \frac{\mathrm{d}x}{x}$ and then extended into the complex plane by analytic continuation. Moreover the functions $\exp$ and $\log$ defined in this way have particularly simple Taylor series (the former is universally convergent) that make their definitions relatively easy to broaden to object other numbers such as matrices, operators and so forth.

What is the probability of this happening? It's precisely equal to 1.0: for that is how we define the natural logarithm (i.e. as the one defined above as opposed to a logarithm with another base or even indeed another function altogether).

This is a pretty vague question, but I take it that you're groping for some "physical significance". The clearest one is that the logarithm is the inverse of the exponential function $x\mapsto e^x$ which itself arises whenever the rate of quantities variation is equal to or proportional to that quantity, a fairly common statement describing physical processes: four example rates of chemical reactions, radio active decays, attenuation of light or other EM radiation through mediums. Given this "physical definition" it follows then that the inverse function is simply that given by $x\mapsto \int_1^x \frac{\mathrm{d}z}{z}$ and then this definition is broadened into the punctured complex plane $\mathbb{C}\sim \{0\}$ by analytic continuation. Moreover the functions $\exp$ and $\log$ defined in this way have particularly simple Taylor series (the former is universally convergent, the latter convergent in an open unit radius circle about $z=1$) that make their definitions relatively easy to broaden to object other numbers such as matrices, operators and so forth.

What is the probability of this happening? It's precisely equal to unity: for that is how we define the natural logarithm (i.e. as the one defined above as opposed to a logarithm with another base or even indeed another function altogether).