# Statistical Thermodynamics and Rate Theories/Essential mathematics for statistical thermodynamics

## Exponentials and Logarithms

Exponential functions in Statistical Thermodynamics play an important role. Most of the functions that are used in this area follow a gaussian distribution. Meaning, a normalized exponential function. Therefore, it is important the understanding of some of the basic properties of this type of functions.

In mathematics, exponential functions are in the form of:

${\displaystyle f(x)=b^{x}}$

Where ${\displaystyle b}$  is any constant, and ${\displaystyle x}$  appears as an exponent. This functions have the particularity to have a close relationship between its derivative and the function itself, due to direct relationship between both of them. However, probably the most relevant exponential function for the Statistical Thermodynamics has to be the exponential function with a base equal to Euler number (${\displaystyle e\approx 2.71828...}$ ), which is the only constant of proportionality between its derivative and the function is 1.

On the other hand, exponential functions have logarithms as their inverse function, having the form of:

${\displaystyle f(x)=\log _{b}(x)}$

This function, also has a particular a relevant significance due to its mathematical properties. It is important to highlight that, along with the exponential function with base , logarithms with base ${\displaystyle e}$  are called natural logarithms.

Some of the properties of the exponential functions are:

• ${\displaystyle e^{x+y}=e^{x}e^{y}\,}$
• ${\displaystyle e^{0}=1\,}$
• ${\displaystyle e^{x}\neq 0}$
• ${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}e^{x}=e^{x}}$
• ${\displaystyle \left(e^{x}\right)^{n}=e^{nx},n\in \mathbb {Z} }$

Some of the properties of logarithm functions are:

• ${\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y}$
• ${\displaystyle \log _{b}\!{\frac {x}{y}}=\log _{b}x-\log _{b}y}$
• ${\displaystyle \log _{b}\left(x^{p}\right)=p\log _{b}x}$
• ${\displaystyle \log _{b}{\sqrt[{p}]{x}}={\frac {\log _{b}x}{p}}}$