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Chapter 1 | Functions and Graphs
1.5 | Exponential and Logarithmic Functions Learning Objectives 1.5.1 Identify the form of an exponential function. 1.5.2 Explain the difference between the graphs of x b and b x . 1.5.3 Recognize the significance of the number e . 1.5.4 Identify the form of a logarithmic function. 1.5.5 Explain the relationship between exponential and logarithmic functions. 1.5.6 Describe how to calculate a logarithm to a different base. 1.5.7 Identify the hyperbolic functions, their graphs, and basic identities.
In this section we examine exponential and logarithmic functions. We use the properties of these functions to solve equations involving exponential or logarithmic terms, and we study the meaning and importance of the number e . We also define hyperbolic and inverse hyperbolic functions, which involve combinations of exponential and logarithmic functions. (Note that we present alternative definitions of exponential and logarithmic functions in the chapter Applications of Integrations , and prove that the functions have the same properties with either definition.) Exponential Functions Exponential functions arise in many applications. One common example is population growth. For example, if a population starts with P 0 individuals and then grows at an annual rate of 2 % , its population after 1 year is P (1) = P 0 +0.02 P 0 = P 0 (1+0.02) = P 0 (1.02). Its population after 2 years is P (2) = P (1)+0.02 P (1) = P (1)(1.02) = P 0 (1.02) 2 . In general, its population after t years is P ( t ) = P 0 (1.02) t , which is an exponential function. More generally, any function of the form f ( x ) = b x , where b >0, b ≠1, is an exponential function with base b and exponent x . Exponential functions have constant bases and variable exponents. Note that a function of the form f ( x ) = x b for some constant b is not an exponential function but a power function. To see the difference between an exponential function and a power function, we compare the functions y = x 2 and y =2 x . In Table 1.10 , we see that both 2 x and x 2 approach infinity as x →∞. Eventually, however, 2 x becomes larger than x 2 and grows more rapidly as x →∞. In the opposite direction, as x →−∞, x 2 →∞, whereas 2 x →0. The line y =0 is a horizontal asymptote for y =2 x .
−3
0 1 2 3 4
5
6
−2
−1
x
x 2
9
0149162536
4
1
2 x
1/81/41/21248163264
Table 1.10 Values of x 2 and 2 x
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