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