# 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 𝑥 1 𝑡 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 𝑑ln(x) 𝑑𝑥 = 1 𝑥 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 𝑑ln(x) 𝑑𝑥 = 1 𝑥 >0 2. The graph of ln(x) is concave downwards, since 𝑑 2 ln(x) 𝑑 𝑥 2 = −1 𝑥 2 <0

lim 𝑥→0 ln 𝑥 =−∞ lim 𝑥→∞ ln 𝑥 =∞
This is consistent with what we know about the graph of ln(x) lim 𝑥→∞ ln 𝑥 =∞

THE DERIVATIVE OF THE NATURAL LOGARITHM AND THE CHAIN RULE
𝑑𝑙𝑛 𝑔 𝑥 𝑑𝑥 = 1 𝑔(𝑥) 𝑔′(𝑥)

ANTIDERIVATIVES INVOLVING THE NATURAL LOGARITHM
Theorem: Remember the domain of the natural log is positive real numbers. 𝑑ln 𝑥 𝑑𝑥 = 1 𝑥 1 𝑥 𝑑𝑥=𝑙𝑛 𝑥 +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 ′.

Example: Differentiate y=ln(3x2-2)3
Rewrite: y=3ln(3x2-2) 𝑦 ′ =3 𝑑𝑙𝑛(3 𝑥 2 −2) 𝑑𝑥 𝑦 ′ = 1 𝑔(𝑥) 𝑔′(𝑥) 𝑦 ′ =3 1 (3 𝑥 2 −2) (6x) 𝑦 ′ = 18𝑥 (3 𝑥 2 −2)

ANTIDERIVATIVES OF SOME TRIGONOMETRIC FUNCTIONS
Memorize these 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