Presentation on theme: "The Natural Logarithmic Function"— Presentation transcript:
1 The Natural Logarithmic Function Section 5.1The Natural Logarithmic Function
2 THE NATURAL LOGARITHMIC FUNCTION Definition: The natural logarithmic function is the function defined byRemember this from thegraphing activity
3 THE DERIVATIVE OF THE NATURAL LOGARITHMIC FUNCTION From the Fundamental Theorem of Calculus, Part 1, we see thatRemember we discussedthis in class
4 LAWS OF LOGARITHMS Remember these rules for logarithms. If x and y are positive numbers and r is a rational number, then
5 PROPERTIES OF THE NATURAL LOGARITHMIC FUNCTION Using calculus, we can describethe natural logarithmic function.Remember x>01. ln x is an increasing function, since2. The graph of ln x is concave downwards, since
6 THEOREMThis is consistent with whatwe know about the graph of ln(x)
7 THE DERIVATIVE OF THE NATURAL LOGARITHM AND THE CHAIN RULE We introduced this in class.
8 ANTIDERIVATIVES INVOLVING THE NATURAL LOGARITHM Theorem:Remember the domain of the natural log is positive real numbers.
9 ANTIDERIVATIVES OF SOME TRIGONOMETRIC FUNCTIONS Memorize these
10 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′.