Properties of Logarithmic Functions

Properties of Logarithmic Functions Logarithmic functions are the inverse of exponential functions where the function is 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: Given logx 16 = 4, find the value of the base x. Solution: The given equation is logx 16 = 4 Convert this Logarithmic equation to Exponential equation by using the formula, logb (a) = N; a = bN Hence logx 16 = 4 can be written as 16 = x4 Now we prime factorization of 16 = 2 * 2 * 2 * 2 Therefore, 16 = 24. This gives 16 = x4; 24 = x4. Hence x = 2. Example 2: For the equation log3 (x2) = 2, then solve for x. Solution: The given equation is log3 (x2) = 2. According to the formula, we have log (am) = m * log a Applying the above formula, we get log3 (x2) = 2; 2 * log3 (x) = 2 Dividing by 2 on both sides; log3 (x) = 2/2; log3 (x) = 1 Now using the formula, logb (a) = N; a = bN. We get, x = 31; x = 3. Hence x = 3 is the solution.