Formula use in Math Olympiad Problem Normally

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 ${\log }_{�}(�)$ and is defined as the exponent to which $�$ must be raised to produce $�$. In other words, ${�}^{{\log }_{�}(�)}=�$.

2. 2.Basic Properties:

   • ${\log }_{�}(1)=0$ for any base $�$.
   • ${\log }_{�}(�)=1$ for any base $�$.

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

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

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

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

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

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: $�=\frac{-�\pm \sqrt{{�}^2-4��}}{2�}$

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: $(�+�{)}^{�}={\sum }_{�=0}^{�}\left(\genfrac{}{}{0px}{}{�}{�}\right){�}^{�-�}{�}^{�}$ 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: $�=\sqrt{�(�-�)(�-�)(�-�)}$

6. Sum of an Arithmetic Series: The sum of the first $�$ terms of an arithmetic series is given by: ${�}_{�}=\frac{�}{2}({�}_1+{�}_{�})$ 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: ${�}_{�}=\frac{{�}_1(1-{�}^{�})}{1-�}$ where ${�}_1$ is the first term, $�$ is the common ratio, and ${�}_{�}$ is the sum.

8. Arithmetic Mean (AM): For

9. 1. $�$ numbers ${�}_1,{�}_2,\dots ,{�}_{�}$