480 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions Natural Logarithms In most practical applications of logarithms, the irrational number e is used as the base. Logarithms with base e are natural logarithms because they occur in the life sciences and economics in natural situations that involve growth and decay. The base e logarithm of x is written ln x (read “el-en x”). The expression ln x represents the exponent to which e must be raised in order to obtain x. Figure 37 2 4 6 8 x –2 2 f(x) = ln x 0 y Figure 35 Natural Logarithm For all positive numbers x, ln x =loge x. A graph of the natural logarithmic function ƒ1x2 = ln x is given in Figure 35. EXAMPLE 5 Evaluating Natural Logarithms with a Calculator Use a calculator to find the values of ln e3, ln 142, and ln 0.005832. SOLUTION Figure 36 shows that the exact value of ln e3 is 3, and that ln 142 ≈4.955827058 and ln 0.005832 ≈ -5.144395284. Figure 36 S Now Try Exercises 45, 51, and 53. Figure 37 illustrates that ln x is the exponent to which e must be raised in order to obtain x. EXAMPLE 6 Measuring the Age of Rocks Geologists sometimes measure the age of rocks by using “atomic clocks.” By measuring the amounts of argon-40 and potassium-40 in a rock, it is possible to find the age t of the specimen in years with the formula t = 11.26 * 1092 ln A1 + 8.33AA KB B ln 2 , where A and K are the numbers of atoms of argon-40 and potassium-40, respectively, in the specimen. (a) How old is a rock in which A = 0 and K70? (b) The ratio A K for a sample of granite from New Hampshire is 0.212. How old is the sample to the nearest hundredth of a billion years? Applications and Models with Natural Logarithms
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