LOGARITHMIC FUNCTIONS By C. SURESH, Mentor in Mathematics, APIIIT-Nuzvid,Krishna(Dt.), A.P., India.

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. Definition: If b≠1 and ‘y’ are any two positive real numbers then there exists a unique real number ‘x’ satisfying the equation b x = y. This x is said to be the logarithm of y to the base b and is written as Log b y = x The idea of logarithms is to reverse the operation of exponentiation.

Thus log 3 9 = 2 since 3 2 = 9 log = 3 since 6 3 = 216 log = -2 since = 0.01 Similarly x 0 = 1 implies that log x 1 = 0 Note: 1.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 to base 2 are called binary logarithms to base ‘e’ are called natural logarithms Identities:

Sol: Given that

FunctionDomainRange exex R (0, ∞ ) log e x (0, ∞ ) R y = log e x y = e x y = x x y (0, 1) (1, 0) Graph:

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