# The Natural Logarithmic Function

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The Natural Logarithmic Function
Section 5.1 The Natural Logarithmic Function

THE NATURAL LOGARITHMIC FUNCTION
Definition: The natural logarithmic function is the function defined by ln(x) = 1 π₯ 1 π‘ dt x>0 Remember this from the graphing activity

THE DERIVATIVE OF THE NATURAL LOGARITHMIC FUNCTION
From the Fundamental Theorem of Calculus, Part 1, we see that πln(x) ππ₯ = 1 π₯ x>0 Remember we discussed this in class

PROPERTIES OF THE NATURAL LOGARITHMIC FUNCTION
Using calculus, we can describe the natural logarithmic function. Remember x>0 1. ln(x) is an increasing function, since πln(x) ππ₯ = 1 π₯ >0 2. The graph of ln(x) is concave downwards, since π 2 ln(x) π π₯ 2 = β1 π₯ 2 <0

lim π₯β0 ln π₯ =ββ lim π₯ββ ln π₯ =β
This is consistent with what we know about the graph of ln(x) lim π₯ββ ln π₯ =β

THE DERIVATIVE OF THE NATURAL LOGARITHM AND THE CHAIN RULE
πππ π π₯ ππ₯ = 1 π(π₯) πβ²(π₯)

ANTIDERIVATIVES INVOLVING THE NATURAL LOGARITHM
Theorem: Remember the domain of the natural log is positive real numbers. πln π₯ ππ₯ = 1 π₯ 1 π₯ ππ₯=ππ π₯ +c HW: p. 329#43-60 x 3 by Tue p. 330#63-87 x 3 by Wed

LOGARITHMIC DIFFERENTIATION
How can we use this information to help us solve problems? Take logarithms of both sides of an equation y = f (x) and use the laws of logarithms to simplify. Differentiate implicitly with respect to x. Solve the resulting equation for y β².

Example: Differentiate y=ln(3x2-2)3
Rewrite: y=3ln(3x2-2) π¦ β² =3 πππ(3 π₯ 2 β2) ππ₯ π¦ β² = 1 π(π₯) πβ²(π₯) π¦ β² =3 1 (3 π₯ 2 β2) (6x) π¦ β² = 18π₯ (3 π₯ 2 β2)

ANTIDERIVATIVES OF SOME TRIGONOMETRIC FUNCTIONS
Memorize these tan π₯ ππ₯=ππ secβ‘(π₯) +πΆ cot π₯ ππ₯=ππ sinβ‘(π₯) +πΆ sec π₯ ππ₯=ππ sec π₯ tanβ‘(π₯) +πΆ csc π₯ ππ₯=ππ csc π₯ cotβ‘(π₯) +πΆ