1. The Natural Logarithmic Function Differentiation

2. Definition of the Natural Logarithmic Function The natural logarithmic function is defined by The domain of the natural logarithmic function is the set of all positive real numbers

3. Properties of the Natural Logarithmic Function The domain is (0, ∞) and the range is (- ∞, ∞). The function is continuous, increasing, and one-to-one. The graph is concave downward.

4. Graph of a the Natural Logarithmic Function

5. Logarithmic Properties If a and b are positive numbers and n is rational, then the following properties are true. 1. ln (1) = 0 2. ln(ab) = ln a + ln b 3. ln(a n ) = n ln a 4. ln (a/b) = ln a – ln b

6. Properties of Logarithms Use the properties of logarithms to approximate ln 0.25 given that ln 2 ≈ 0.6931 and ln 3 ≈ 1.0986 (b) ln 24 (c) ln 1/72

7. Expanding Logarithmic Expressions Use the properties of logarithms to expand the logarithmic expression

8. Logarithms as a Single Quantity Write the expression as a logarithm of a single quantity (a) 3 ln x + 2 ln y – 4 ln z (b) 2 ln 3 - ½ln (x 2 + 1) (c) ½[ln (x 2 + 1) – ln (x + 1) – ln (x – 1)]

9. The Number e The base of the natural logarithmic function is e e ≈ 2.71828182846...

10. Definition of e The letter e denotes the positive real number such that

11. Evaluating Natural Logarithmic Expressions ln2 ln 32 ln 0.1

12. Derivative of the Natural Logarithmic Function In other words, the derivative of the function over the function.

13. Differentiation of Logarithmic Functions Find the derivative of the function (a) h(x) = ln (2x 2 + 1) (b) f(x) = x ln x

14. Differentiation of Logarithmic Functions

15. Logarithmic Properties as Aids to Differentiation

17. More Examples P. 322 problems 60 On-line Examples

18. Logarithmic Differentiation

20. P. 322 problems 87 – 92 On-line Examples

21. Finding the Equation of the Tangent Line Find an equation of the tangent line to the graph of f at the indicated point

22. Locating Relative Extrema Locate any relative extrema and inflection points for the graph of Y = x – ln x Y = lnx/x