1. LOGARITHMIC FUNCTIONS

2. 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.

3. Thus log 3 9 = 2 since 3 2 = 9 log 6 216 = 3 since 6 3 = 216 log 10 0.01 = -2 since 10 -2 = 0.01 Similarly x 0 = 1 implies that log x 1 = 0 Notes: 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.

4. 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:

5. Sol: Given that

7. 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: