83 I 10 83 I R I I R log I I 10 83 I I 10 83 10 83 I R log

83 i 10 83 i r i i r log i i 10 83 i i 10 83 10 83 i

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= 8.3 = log 10 8.3 I . 10 8.3 I 0 R = log 10 8.3 I 0 I 0 R = log I I 0 10 8.3 I 0 . I , 10 8.3 10 8.3 I 0 R = log I I 0 , I R , EXAMPLE 9 Use natural logarithms. [ 10, 10, 1] by [ 10, 10, 1] Figure 3.14 The domain of is 1 - q , 3 2 . f 1 x 2 = ln 1 3 - x 2 [ 10, 10, 1] by [ 10, 10, 1] Figure 3.15 3 is excluded from the domain of h 1 x 2 = ln 1 x - 3 2 2 . 408 Chapter 3 Exponential and Logarithmic Functions
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Section 3.2 Logarithmic Functions 409 The basic properties of logarithms that were listed earlier in this section can be applied to natural logarithms. Properties of Natural Logarithms Inverse properties General Properties Natural Logarithms log b 1=0 1. log b b=1 2. log b b x =x 3. b log b x =x 4. ln 1=0 1. ln e=1 2. ln e x =x 3. e ln x =x 4. Examine the inverse properties, and Can you see how ln and “undo” one another? For example, Dangerous Heat: Temperature in an Enclosed Vehicle When the outside air temperature is anywhere from 72° to 96° Fahrenheit, the temperature in an enclosed vehicle climbs by 43° in the first hour. The bar graph in Figure 3.16 shows the temperature increase throughout the hour.The function models the temperature increase, in degrees Fahrenheit, after minutes. Use the function to find the temperature increase, to the nearest degree, after 50 minutes. How well does the function model the actual increase shown in Figure 3.16 ? x f 1 x 2 , f 1 x 2 = 13.4 ln x - 11.6 EXAMPLE 11 ln e 2 = 2, ln e 7 x 2 = 7 x 2 , e ln 2 = 2, and e ln 7 x 2 = 7 x 2 . e e ln x = x . ln e x = x 60 43 ° 50 41 ° 40 38 ° 30 34 ° 20 29 ° 45 ° 40 ° 35 ° 30 ° 25 ° 20 ° 15 ° 10 ° Temperature Increase ( ° F) Temperature Increase in an Enclosed Vehicle Minutes 10 19 ° 5 ° Figure 3.16 Source: Professor Jan Null, San Francisco State University Solution We find the temperature increase after 50 minutes by substituting 50 for and evaluating the function at 50. This is the given function. Substitute 50 for Graphing calculator keystrokes: On some calculators, a parenthesis is needed after 50. According to the function, the temperature will increase by approximately 41° after 50 minutes. Because the increase shown in Figure 3.16 is 41°, the function models the actual increase extremely well. Check Point 11 Use the function in Example 11 to find the temperature increase, to the nearest degree, after 30 minutes. How well does the function model the actual increase shown in Figure 3.16 ? 13.4 ln 50 - 11.6 ENTER . L 41 x . f 1 50 2 = 13.4 ln 50 - 11.6 f 1 x 2 = 13.4 ln x - 11.6 x
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The Curious Number e You will learn more about each curiosity mentioned below when you take calculus. The number was named by the Swiss mathematician Leonhard Euler (1707–1783), who proved that it is the limit as of features in Euler’s remarkable relationship in which The first few decimal places of are fairly easy to remember: The best rational approximation of using numbers less than 1000 is also easy to remember: Isaac Newton (1642–1727), one of the cofounders of calculus, showed that from which we obtain an infinite sum suitable for calculation because its terms decrease so rapidly. ( Note : ( factorial) is the product of all the consecutive integers from down to 1: ) The area of the region bounded by the and
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  • Fall '16
  • Kirkpatrick
  • Physics, pH, Natural logarithm, Logarithm, Richter scale

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