# Section 7.2* - Gordon State College Section 7.3 The Natural Logarithmic Function THE NATURAL LOGARITHMIC FUNCTION Definition: The natural logarithmic function is the function defined by x 1

ln x dt 1 t x 0 THE DERIVATIVE OF THE NATURAL LOGARITHMIC FUNCTION From the Fundamental Theorem of Calculus, Part 1, we see that d

1 (ln x) dx x LAWS OF LOGARITHMS If x and y are positive numbers and r is a rational number, then 1. ln( xy ) ln x ln y x

2. ln ln x ln y y r 3. ln x r ln x PROPERTIES OF THE NATURAL LOGARITHMIC FUNCTION 1. ln x is an increasing function, since d

1 (ln x) 0 dx x 2. The graph of ln x is concave downwards, since 2 d 1 (ln x) 2 0 2

dx x THEOREM 1. lim (ln x) x 2. lim (ln x) x 0 THE NUMBER e

Definition: e is the number such that ln e = 1 e 2.718281828459045 . . . e 2.7 1828 1828 45 90 45 . . . THE DERIVATIVE OF THE NATURAL LOGARITHM AND THE CHAIN RULE d g ( x) ln g ( x)

dx g ( x) ANTIDERIVATIVES INVOLVING THE NATURAL LOGARITHM Theorem: d 1 ln | x | dx

x 1 dx ln | x | C x ANTIDERIVATIVES OF SOME TRIGONOMETRIC FUNCTIONS tan x dx

ln | sec x | C cot x dx ln | sin x | C sec x dx ln | sec x tan x | C

csc x dx ln | csc x cot x | C LOGARITHMIC DIFFERENTIATION 1. Take logarithms of both sides of an equation y = f (x) and use the laws of logarithms to simplify. 2. Differentiate implicitly with respect to x. 3. Solve the resulting equation for y.