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The Natural Logarithmic Function

The Natural Logarithmic Function. Differentiation. 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. Properties of the Natural Logarithmic Function.

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The Natural Logarithmic Function

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  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(an) = 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 (x2 + 1) • (c) ½[ln (x2 + 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 (2x2 + 1) • (b) f(x) = x ln x

  14. Differentiation of Logarithmic Functions

  15. Logarithmic Properties as Aids to Differentiation

  16. Logarithmic Properties as Aids to Differentiation

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

  18. Logarithmic Differentiation

  19. Logarithmic Differentiation

  20. Logarithmic Differentiation • 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

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