exp_log - Exponential and logarithm functions The graph of...

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Exponential and logarithm functions The graph of = y 2 x Let f be defined by = ( ) f x 2 x . We compute some values of f in order to sketch its graph. = ( ) f 1 2 ( ) 1 = 2, = ( ) f 2 2 2 = 4, = ( ) f 3 2 3 = 8, = ( ) f 0 2 0 = 1, = ( ) f - 1 2 ( ) - 1 = 1 2 = 0.5, = ( ) f - 2 2 ( ) - 2 = 1 2 2 = 1 4 = 0.25, = ( ) f - 3 2 ( ) - 3 = 1 2 3 = 1 8 = 0.125, = f 1 2 2 1 2 = 2 ~ 1.41421, = f 3 2 2 . 2 1 2 = 2 2 ~ 2.82843, = f 5 2 2 5 2 = 2 2 . 2 1 2 = 4 2 ~ 5.65685, = f - 1 2 2 - 1 2 = 1 2 1 2 = 1 2 = 2 2 ~ 0.707107, = f - 3 2 2 - 3 2 = 1 2 3 2 = 1 2 2 = 2 4 ~ 0.353553, = f - 5 2 2 - 5 2 = 1 2 5 2 = 1 4 2 = 2 8 ~ 0.176777 These values are collected together in the following table. x | - 3 - 5 2 - 2 - 3 2 - 1 - 1 2 0 1 2 1 3 2 2 5 2 3 ( ) f x | 1 8 2 8 1 4 2 4 1 2 2 2 1 2 2 2 2 4 4 2 8 | 0 .125000 0 .176777 0 .250000 0 .353553 0 .500000 0 .707107 1 1.41421 2 2.82843 4 5.65685 8 Joining the points given by the table to form a continuous (unbroken) curve amounts to suggesting that the domain of the function f is the set of all real numbers.
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In order to justify this first consider what 2 x means if x is a rational number . For a positive rational number = x p q , where p and q are positive integers, = ( ) f x 2 p q = q 2 p , that is, the q th root of 2 p . For example, = ( ) f 1.6 = f 8 5 2 8 5 = 5 2 8 = 5 256 ~ 3.03143. For a negative rational number = x - p q , where p is a negative integer and q is a positive integer, = ( ) f x 2 - p q = 1 2 p q = 1/ q 2 p . For example, = ( ) f - 1.6 = f - 8 5 2 - 8 5 = 1/ 5 2 8 ~ 0.329877. For a general real number x we can think of x being being approximated sufficiently accurately by a rational number in order to calculate = ( ) f x 2 x , to a desired degree of accuracy. For example, we can imagine 2 π being computed to 6 figure accuracy by using a 7 figure rational approximation for π , namely = 3.141593 3141593 1000000 . Then 2 π ~ 1000000 2 3141593 ~ 8.82498. Actually, this is a very inefficient way to perform the computation if all the digits of 2 3141593 are obtained since there are 945714 of them. However, this example is given in an attempt to explain why the domain of the function f where = ( ) f x 2 x is indeed the set of
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exp_log - Exponential and logarithm functions The graph of...

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