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 Remember this from the graphing activity

THE DERIVATIVE OF THE NATURAL LOGARITHMIC FUNCTION From the Fundamental Theorem of Calculus, Part 1, we see that Remember we discussed this in class

LAWS OF LOGARITHMS Remember these rules for logarithms. If x and y are positive numbers and r is a rational number, then

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 2. The graph of ln x is concave downwards, since

THEOREM This is consistent with what we know about the graph of ln(x)

THE DERIVATIVE OF THE NATURAL LOGARITHM AND THE CHAIN RULE We introduced this in class.

ANTIDERIVATIVES INVOLVING THE NATURAL LOGARITHM Theorem: Remember the domain of the natural log is positive real numbers.

ANTIDERIVATIVES OF SOME TRIGONOMETRIC FUNCTIONS Memorize these

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 ln(3x2-2)

Example: Differentiate y=ln(3x2-2)3 Rewite: y=3ln(3x2-2) y’ = 3 ln(3x2-2)