SECTION 5.3 Exponential Functions 295 Using a Calculator to Evaluate Powers of 2 Use a calculator to evaluate: (a) 21.4 (b) 21.41 (c) 21.414 (d) 21.4142 (e) 2 2 Solution EXAMPLE 1 Figure 18 shows the solutions for parts (a) and (e) using a TI-84 Plus CE graphing calculator. (a) ≈ 2 2.639015822 1.4 (b) ≈ 2 2.657371628 1.41 (c) ≈ 2 2.66474965 1.414 (d) ≈ 2 2.665119089 1.4142 (e) ≈ 2 2.665144143 2 Figure 18 x f x( ) 0 5 1 10 2 20 3 40 4 80 Table 1 But what is the meaning of a ,x where the base a is a positive real number and the exponent x is an irrational number? Although a rigorous definition requires methods discussed in calculus, the basis for the definition is easy to follow: Select a rational number r that is formed by truncating (removing) all but a finite number of digits from the irrational number x. Then it is reasonable to expect that ≈ a a x r For example, take the irrational number π = 3.14159 . . . . Then an approximation to πa is ≈ πa a3.14 where the digits of π after the hundredths position are truncated.A better approximation is ≈ πa a3.14159 where the digits after the hundred-thousandths position are truncated. Continuing in this way, we can obtain approximations to πa to any desired degree of accuracy. Most calculators have an xy key or a caret key ∧ for working with exponents. To evaluate expressions of the form a ,x enter the base a, then press the xy key (or the ∧ key), enter the exponent x, and press = (or ENTER ). Now Work PROBLEM 19 THEOREM Laws of Exponents If s t a , , ,and b are real numbers with > a 0 and > b 0, then • ⋅ = + a a a s t s t • ( ) = a a s t st • ( ) = ⋅ ab a b s s s • 1 1 s = • ( ) = = −a a a 1 1 s s s • = a 1 0 (1) Introduction to Exponential Growth Suppose a function f has the following two properties: • The value of f doubles with every 1-unit increase in the independent variable x. • The value of f at = x 0 is 5, so ( ) = f 0 5. Table 1 shows values of the function f for = x 0, 1, 2, 3, and 4. It can be shown that the laws for rational exponents hold for real exponents.
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