In mathematics, particularly in math Olympiads, logarithms are often used to solve various types of problems. Here are some commonly used logarithm formulas:

1.Definition: The logarithm of a number $\ufffd$ to the base $\ufffd$ is denoted by ${\mathrm{log}}_{\ufffd}(\ufffd)$ and is defined as the exponent to which $\ufffd$ must be raised to produce $\ufffd$. In other words, ${\ufffd}^{{\mathrm{log}}_{\ufffd}(\ufffd)}=\ufffd$.

2.Basic Properties:

- ${\mathrm{log}}_{\ufffd}(1)=0$ for any base $\ufffd$.
- ${\mathrm{log}}_{\ufffd}(\ufffd)=1$ for any base $\ufffd$.

3.Change of Base Formula: For any positive numbers $\ufffd$, $\ufffd$, and $\ufffd$ where $\ufffd\mathrm{\ne}1$, we have: ${\mathrm{log}}_{\ufffd}(\ufffd)=\frac{{\mathrm{log}}_{\ufffd}(\ufffd)}{{\mathrm{log}}_{\ufffd}(\ufffd)}$

4.Product Rule: ${\mathrm{log}}_{\ufffd}(\ufffd\ufffd)={\mathrm{log}}_{\ufffd}(\ufffd)+{\mathrm{log}}_{\ufffd}(\ufffd)$ for all positive $\ufffd$ and $\ufffd$.

5.Quotient Rule: ${\mathrm{log}}_{\ufffd}\left(\frac{\ufffd}{\ufffd}\right)={\mathrm{log}}_{\ufffd}(\ufffd)-{\mathrm{log}}_{\ufffd}(\ufffd)$ for all positive $\ufffd$ and $\ufffd$.

6.Power Rule: ${\mathrm{log}}_{\ufffd}({\ufffd}^{\ufffd})=\ufffd{\mathrm{log}}_{\ufffd}(\ufffd)$ for all positive $\ufffd$ and real $\ufffd$.

7.Change of Base Formula for Natural Logarithm: $\mathrm{ln}(\ufffd)=\frac{\mathrm{log}(\ufffd)}{\mathrm{log}(\ufffd)}=\frac{\mathrm{log}(\ufffd)}{1}=\mathrm{log}(\ufffd)$

These are some of the fundamental logarithm formulas used in mathematical Olympiads.

Quadratic Formula: For a quadratic equation $\ufffd{\ufffd}^{2}+\ufffd\ufffd+\ufffd=0$, the solutions are given by: $\ufffd=\frac{-\ufffd\pm \sqrt{{\ufffd}^{2}-4\ufffd\ufffd}}{2\ufffd}$

Vieta's Formulas: For a quadratic equation $\ufffd{\ufffd}^{2}+\ufffd\ufffd+\ufffd=0$ with roots ${\ufffd}_{1}$ and ${\ufffd}_{2}$, the sum of roots is $-\frac{\ufffd}{\ufffd}$ and the product of roots is $\frac{\ufffd}{\ufffd}$.

Binomial Theorem: $(\ufffd+\ufffd{)}^{\ufffd}={\sum}_{\ufffd=0}^{\ufffd}\left(\genfrac{}{}{0px}{}{\ufffd}{\ufffd}\right){\ufffd}^{\ufffd-\ufffd}{\ufffd}^{\ufffd}$ where $\left(\genfrac{}{}{0px}{}{\ufffd}{\ufffd}\right)$ is the binomial coefficient.

Pythagorean Theorem: In a right-angled triangle, the square of the length of the hypotenuse ($\ufffd$) is equal to the sum of the squares of the other two sides ($\ufffd$ and $\ufffd$): ${\ufffd}^{2}={\ufffd}^{2}+{\ufffd}^{2}$.

Heron's Formula: For a triangle with sides of lengths $\ufffd$, $\ufffd$, and $\ufffd$, and semi-perimeter $\ufffd$, the area $\ufffd$ is given by: $\ufffd=\sqrt{\ufffd(\ufffd-\ufffd)(\ufffd-\ufffd)(\ufffd-\ufffd)}$

Sum of an Arithmetic Series: The sum of the first $\ufffd$ terms of an arithmetic series is given by: ${\ufffd}_{\ufffd}=\frac{\ufffd}{2}({\ufffd}_{1}+{\ufffd}_{\ufffd})$ where ${\ufffd}_{1}$ is the first term, ${\ufffd}_{\ufffd}$ is the $\ufffd$th term, and ${\ufffd}_{\ufffd}$ is the sum.

Sum of a Geometric Series: The sum of the first $\ufffd$ terms of a geometric series is given by: ${\ufffd}_{\ufffd}=\frac{{\ufffd}_{1}(1-{\ufffd}^{\ufffd})}{1-\ufffd}$ where ${\ufffd}_{1}$ is the first term, $\ufffd$ is the common ratio, and ${\ufffd}_{\ufffd}$ is the sum.

Arithmetic Mean (AM): For

$\ufffd$ numbers ${\ufffd}_{1},{\ufffd}_{2},\dots ,{\ufffd}_{\ufffd}$