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