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Once again, you can take advantage of the rule for converting a power to a product This time, you have log10 e = e log10
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Consider equal to 314159, and consider e equal to 271828 Then log10 e 271828 log10 314159 271828 049715 135139 When you take the common antilog of this, you ll get e because antilog10 (log10 e) = e Calculating, you should get antilog10 135139 2245898 When I input the numbers into my calculator and use the x ^y key, I get 314159271828 2245906
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Rounding error strikes again! And as before, if you wish, you can see for yourself how this error shrinks to the vanishing point if you let your calculator keep all of its extra digits until the final step
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In most real-life applications of exponentials, the base b is either 10 or e However, once in a while you ll come across a situation where b is some other positive real number This section describes the basic properties that hold for exponentials in general
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Reciprocal vs negative exponent Suppose that x is some real number The reciprocal of the exponential of x is equal to the exponential of the negative of x, as follows:
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1 / (b x) = b x when b > 0 You should recognize this from your work with powers and roots Here s a familiar example You know that 1/8 is equal to 1 / (23) This is the same as saying that 1/8 is equal to 2 3 You also know that 1/100 equals 1 / (102), which is the same as saying that 1/100 equals 10 2 Now consider this, rounded to four decimal places: 1 / (e 3) 1 / (27183) 1 / 20079 00498
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Compare the above with the result of entering 3 into a scientific calculator, then hitting Inv, then hitting ln, and finally rounding to four decimal places: e 3 00498 Rounding error stayed out of this little exercise!
Product vs sum Exponential functions can express the relationship between sums and products, just as logarithms do Suppose that x and y are real numbers Then
b xb y = b (x+y) when b > 0 To demonstrate, let b = 10, x = 4, and y = 6 We can plug in the numbers for the product of exponentials and get and get 104 10 6 = 10,000 0000001 = 001 When we evaluate the right side, we get 10[4+( 6)] = 10(4 6) = 10 2 = 001 The results agree You ll find that this is always true, no matter what base and arguments you use, as long as the base is positive Of course, if you get nonterminating decimals for any of the values in the calculation, you should expect some rounding error
Ratio vs difference Again, suppose that x and y are real numbers Then
b x / b y = b(x y) when b > 0 Using the same numerical values as before, we can demonstrate this We plug in the numbers on the left side of the equation and get 104 / 10 6 = 10,000 / 0000001 = 10,000 1,000,000 = 10,000,000,000 = 1010 Then we can evaluate the right side to see that 10[4 ( 6)] = 10(4+6) = 1010
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Ratio in an exponent Here s a more complicated property of exponentials Let x and y be real numbers, with the restriction that y cannot be equal to 0 Then
b(x /y) = (b x)(1/y) when b > 0 Let s try an example where the base b is 10, with exponents x = 4 and y = 7 Evaluating the left side first, letting 4/7 05714 and using the x y or x ^y function key on a calculator, we get 10(4/7) 1005714 3727 Alternatively, we can enter 05714, hit the Inv key, and then hit log to find 10 to the power of 05714 Now when we plug the numbers into the right side of the general equation and work it out, we obtain (104)(1/7) = 10,000(1/7) To work this out on a calculator, we must first figure 1/7 to four decimal places That gives us 01429 Then, we enter 10,000, hit the x y or x^y key, and enter 01429 The result is 3729 There s a discrepancy, because we ve taken a rounding error to the seventh power!
Power of a power vs product Exponentials can show the relationship between a power of a power and a product Suppose that x and y are real numbers Then
(b x)y = b(xy) when b > 0 To demonstrate this, let b = e, x = 2, and y = 3 Let s evaluate the left side first, using 2718 as the value of e and going to three decimal places during the calculation process: (e 2)3 (27182)3 73883 403256 Now the right side: e(2 3) = e6 27186 403178
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