Properties of Logarithms

Properties of Logarithms Logarithmic are the inverse of exponential functions. They are commonly used in many branches in math. Logarithms are written as f(x) = log b (a) such that b 0, b = 1 and a 0. This is read as log base b of a. Logarithmic functions have many properties and rule which are used to solve many questions: General properties (where x 0, y0) logb (xy) = logbx + logb y logb (x/y) = logbx - logby logb (xm) = m logb x logb b = 1 Example 1: Solve logx 27 = 3, find the value of the base x. Solution: The given equation is logx 27 = 3 Convert this Logarithmic equation to Exponential equation by using the formula, logb (a) = N; a = bN Hence logx 16 = 4 can be written as 27 = x3 Now we prime factorization of 27 = 3 * 3 * 3. Therefore, 27 = 33. This gives 27 = x3; 33 = x3. Hence x = 3 is the solution. Example 2: Solve logx 225 = 2, find the value of the base x. Solution: The given equation is logx 225 = 2. Convert this Logarithmic equation to Exponential equation by using the formula, logb (a) = N; a = bN Hence logx 225 = 2 can be written as 225 = x2. Now we prime factorization of 225 = 3 * 3 * 5 * 5. Therefore, 225 = 152. This gives 225 = x2; 152 = x2. Hence x = 15 is the solution.