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Understanding Natural Logarithmic Function

Learn about the natural logarithmic function, its properties, derivative, antiderivatives, and the number 'e'. Explore the laws and theorems associated with ln(x) with practical examples.

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Understanding Natural Logarithmic Function

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  1. Section 7.3 The Natural Logarithmic Function

  2. THE NATURAL LOGARITHMIC FUNCTION Definition: The natural logarithmic function is the function defined by

  3. THE DERIVATIVE OF THE NATURAL LOGARITHMIC FUNCTION From the Fundamental Theorem of Calculus, Part 1, we see that

  4. LAWS OF LOGARITHMS If x and y are positive numbers and r is a rational number, then

  5. PROPERTIES OF THE NATURAL LOGARITHMIC FUNCTION 1. ln x is an increasing function, since 2. The graph of ln x is concave downwards, since

  6. THEOREM

  7. THE NUMBER e Definition:e is the number such that ln e = 1 e≈ 2.718281828459045 . . . e ≈ 2.7 1828 1828 45 90 45 . . .

  8. THE DERIVATIVE OF THE NATURAL LOGARITHM AND THE CHAIN RULE

  9. ANTIDERIVATIVES INVOLVING THE NATURAL LOGARITHM Theorem:

  10. ANTIDERIVATIVES OF SOME TRIGONOMETRIC FUNCTIONS

  11. LOGARITHMIC DIFFERENTIATION • 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′.

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