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

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

2. 2.Basic Properties:

• ${\mathrm{log}}_{�}\left(1\right)=0$ for any base $�$.
• ${\mathrm{log}}_{�}\left(�\right)=1$ for any base $�$.
3. 3.Change of Base Formula: For any positive numbers $�$, $�$, and $�$ where $�\mathrm{\ne }1$, we have: ${\mathrm{log}}_{�}\left(�\right)=\frac{{\mathrm{log}}_{�}\left(�\right)}{{\mathrm{log}}_{�}\left(�\right)}$

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

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

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

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

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

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

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

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

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

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

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

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

8. Arithmetic Mean (AM): For

1. $�$ numbers ${�}_{1},{�}_{2},\dots ,{�}_{�}$