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# lecture23 - Math 121 Lecture 23 4.4 The Natural Logarithm...

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§ 4.4 The Natural Logarithm Function Math 121 Lecture 23 ! The Natural Logarithm of x ! Properties of the Natural Logarithm ! Exponential Expressions ! Solving Exponential Equations ! Solving Logarithmic Equations ! Other Exponential and Logarithmic Functions ! Common Logarithms ! Max’s and Min’s of Exponential Equations

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Definition Example Natural logarithm of x : Given the graph of y = e x , the reflection of that graph about the line y = x , denoted y = ln x
1) The point (1, 0) is on the graph of y = ln x [because (0, 1) is on the graph of y = e x ]. 2) ln x is defined only for positive values of x . 3) ln x is negative for x between 0 and 1. 4) ln x is positive for x greater than 1. 5) ln x is an increasing function and concave down. Simplify. Using properties of the exponential function, we have e ln3 " 2ln x = e ln3 e 2ln x = 3 e 2 # ln x = 3 ( e ln x ) 2 = 3 x 2 .

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Solve the equation for x . e
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lecture23 - Math 121 Lecture 23 4.4 The Natural Logarithm...

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