Exponentiation: The third power of some number ‘b’ is the product of 3 factors of ‘b’. More generally, raising ‘b’ to the n-th power (n is a natural number) is done by multiplying n factors. The idea of logarithms is to reverse the operation of exponentiation. Definition: If b≠1 and ‘y’ are any two positive real numbers then there exists a unique real number ‘x’ satisfying the equation bx = y. This x is said to be the logarithm of y to the base b and is written as Logb y = x
Thus log3 9 = 2 since 32 = 9 log6 216 = 3 since 63 = 216 log10 0.01 = -2 since 10-2 = 0.01 Similarly x0 = 1 implies that logx 1 = 0 Note: • Since the exponential function value can never be zero, we can say that logarithm of zero is undefined. 2. Similarly, logarithmic function is not defined for negative values.
Types of logarithms: • logarithms to base 10 are called common logarithms • logarithms to base 2 are called binary logarithms • logarithms to base ‘e’ are called natural logarithms • Identities:
y Graph: y = x (0 , 1) y = ex y = loge x x (1 , 0)