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 1, we have: log()=log()log()

  4. 4.Product Rule: log()=log()+log() for all positive and .

  5. 5.Quotient Rule: log()=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()=log()log()=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: =±242

  2. Vieta's Formulas: For a quadratic equation 2++=0 with roots 1 and 2, the sum of roots is and the product of roots is .

  3. Binomial Theorem: (+)==0() where () 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: =()()()

  6. Sum of an Arithmetic Series: The sum of the first terms of an arithmetic series is given by: =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: =1(1)1 where 1 is the first term, is the common ratio, and is the sum.

  8. Arithmetic Mean (AM): For

    1. numbers 1,2,,