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 ${�}_1,{�}_2,\dots ,{�}_{�}$, the arithmetic mean is given by: $\text{AM}=\frac{{�}_1+{�}_2+\dots +{�}_{�}}{�}$

   2. Geometric Mean (GM): For $�$ positive numbers ${�}_1,{�}_2,\dots ,{�}_{�}$, the geometric mean is given by: $\text{GM}=\sqrt[�]{{�}_1\cdot {�}_2\cdot \dots \cdot {�}_{�}}$

   3. Harmonic Mean (HM): For $�$ positive numbers ${�}_1,{�}_2,\dots ,{�}_{�}$, the harmonic mean is given by: $\text{HM}=\frac{�}{\frac{1}{{�}_1}+\frac{1}{{�}_2}+\dots +\frac{1}{{�}_{�}}}$

   4. Quadratic Mean (RMS): For $�$ numbers ${�}_1,{�}_2,\dots ,{�}_{�}$, the quadratic mean is given by: $\text{RMS}=\sqrt{\frac{{�}_1^2+{�}_2^2+\dots +{�}_{�}^2}{�}}$

   5. Sum of Cubes: $1^3+2^3+\dots +{�}^3={\left(\frac{�(�+1)}{2}\right)}^2$

   6. Difference of Cubes: ${�}^3-{�}^3=(�-�)({�}^2+��+{�}^2)$

   7. Pascal's Identity: $\left(\genfrac{}{}{0px}{}{�}{�}\right)=\left(\genfrac{}{}{0px}{}{�-1}{�-1}\right)+\left(\genfrac{}{}{0px}{}{�-1}{�}\right)$

   8. Euler's Formula: For a convex polyhedron with $�$ vertices, $�$ edges, and $�$ faces, the formula holds: $�-�+�=2$.

   9. Volume of a Sphere: The volume of a sphere with radius $�$ is $\frac{4}{3}�{�}^3$.

   10. Surface Area of a Sphere: The surface area of a sphere with radius $�$ is $4�{�}^2$.

   11. Circumference of a Circle: The circumference of a circle with radius $�$ is $2��$.

   12. Area of a Circle: The area of a circle with radius $�$ is $�{�}^2$.

   13. Law of Sines: For a triangle with sides $�$, $�$, and $�$, and opposite angles $�$, $�$, and $�$, the law of sines states: $\frac{�}{\sin �}=\frac{�}{\sin �}=\frac{�}{\sin �}$.

   14. Law of Cosines: For a triangle with sides $�$, $�$, and $�$, and angle $�$ opposite side $�$, the law of cosines states: ${�}^2={�}^2+{�}^2-2��\cos �$.

   15. 2. Sum of an Arithmetic Series: The sum of the first $�$ terms of an arithmetic series $�,�+�,�+2�,\dots$ is given by: ${�}_{�}=\frac{�}{2}(2�+(�-1)�)$

       3. Sum of a Geometric Series: The sum of the first $�$ terms of a geometric series $�,��,�{�}^2,\dots$ is given by: ${�}_{�}=\frac{�(1-{�}^{�})}{1-�}$ (for $�\mathrm{\ne }1$)

       4. Sum of an Infinite Geometric Series: The sum of an infinite geometric series $�,��,�{�}^2,\dots$ with $\mathrm{\mid }�\mathrm{\mid }<1$ is given by: ${�}_{\mathrm{\infty }}=\frac{�}{1-�}$

       5. Binomial Theorem: The expansion of $(�+�{)}^{�}$ is given by: $(�+�{)}^{�}=\left(\genfrac{}{}{0px}{}{�}{0}\right){�}^{�}{�}^0+\left(\genfrac{}{}{0px}{}{�}{1}\right){�}^{�-1}{�}^1+\dots +\left(\genfrac{}{}{0px}{}{�}{�}\right){�}^0{�}^{�}$

       6. Sum of Binomial Coefficients: The sum of the binomial coefficients in the $�$th row of Pascal's Triangle is $2^{�}$.

       7. Fibonacci Sequence: The $�$th term of the Fibonacci sequence $1,1,2,3,5,8,\dots$ is given by: ${�}_{�}={�}_{�-1}+{�}_{�-2}$ with initial conditions ${�}_0=0$ and ${�}_1=1$.

       8. Quadratic Formula: The solutions to the quadratic equation $�{�}^2+��+�=0$ are given by: $�=\frac{-�\pm \sqrt{{�}^2-4��}}{2�}$

       9. Law of Tangents: In a triangle $���$, the law of tangents states: $$�−��+�=tan⁡(12(�−�))tan⁡(12(�+�))