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Notes on the Graph of the Natural Logarithm Function

Notes on the Graph of the Natural Logarithm Function -...

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Notes on the Graph of the Natural Logarithm Function Definition : For 0 x define 1 ln x dt x t = called the natural logarithm function. Notice the domain of this function is the interval ( 29 0, since the function 1 t is continuous on the interval ( 29 0, . When 1 x , ln x represents the area bounded by the curve 1 y t = , the x- axis, and the vertical lines 1 t = and t x = . When 0 1 x < < , 1 1 ln x x dt dt x t t = = - . So in this case ln x represents the negative of the area bounded by the curve 1 y t = , the x-axis, and the vertical lines 1 t = and t x = . Properties of the Graph of the natural logarithm function : A summary of the properties of the graph are listed below for easy reference. The justification is given later. A. The point ( 29 1,0 lies on the graph of the natural logarithm function. B. The function ( 29 ln f x x = is continuous and increasing on the interval ( 29 0, . C. The function ( 29 ln f x x = is concave down on the interval ( 29 0, . From the properties listed above ( 29 2 1 2 2 1 ln 0 d d f x x x dx dx x - ′′ = = = - < . D. limln x x →∞ = ∞ and the graph of the natural logarithm function does not have horizontal asymptote. E. 0 lim ln x x → + = -∞ and the line 0 x = , that is the y-axis, is a vertical asymptote for the natural logarithm function. F. The following figure demonstrates these properties. 1
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Laws of the Natural Logarithm : The natural logarithm function yields the common Laws of logarithms you know from high school. Below they are listed for easy reference. The justification is given later. In the list below the numbers a and b are positive numbers and r is any real number. 1. ln ln ln ab a b = + 2. ln ln r a r a = 3. ln ln ln a a b b = - 4. ln ln a b = if and only if a b = . 5. There is exactly one number, called e , for which ln 1 e = . Below are examples illustrating the use of the above facts to solve problems involving logarithms. Example 1 : Use the Laws of logarithms to expand the quantity 6 2 ln 3 yz x + . Solution : 6 6 2 2 ln ln ln 3 3 yz yz x x = - + + ( Law 3 ) ( 29 1 6 2 2 ln ln ln 3 y z x = + - + ( Law 1 ) ( 29 2 1 ln 6ln ln 3 2 y z x = + - + ( Law 2 ) Example 2 : Express the quantity 1 ln 2 ln8 3ln16 4 + - as a single logarithm.
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