how to use barcode in c#.net If we say that the natural logarithm of a certain number p is equal to q, what do we mean in Software

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If we say that the natural logarithm of a certain number p is equal to q, what do we mean
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Answer 29-4
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The natural logarithm (or natural log) of a number is the power to which we must raise Euler s constant, e, to get that number If we say that the natural log of p is equal to q, we mean p = eq The common log is sometimes called the base-e log, because e is the base that we raise to various powers The value of e is approximately 271828 It s an irrational number, however, so it cannot be fully written out in decimal form
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Question 29-5
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According to the definition in Answer 29-4, what is the natural log of e Of e 2 Of e 3 Of e4 What happens to the natural log of a number as that number grows larger without limit
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Part Three 541 Answer 29-5
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The natural log is related to a growing number like this: The natural log of e is 1, because e 1 = e The natural log of e 2 is 2 The natural log of e 3 3 The natural log of e 4 is 4
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As a number gets larger without limit, so does its natural log, but the size of the log grows much more slowly than the size of the number
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According to the definition in Answer 29-4, What is the natural log of 1 Of 1/e Of 1/e 2 Of 1/e 3 What happens to the natural log of a positive real number whose absolute value keeps shrinking What happens to the natural log of a shrinking positive real number when it actually becomes 0
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Answer 29-6
As the absolute value of a positive number keeps shrinking, its natural log changes like this: The natural log of 1 is 0, because e 0 = 1 The natural log of 1/e is 1 The natural log of 1/e 2 is 2 The natural log of 1/e 3 is 3
As a positive number approaches 0, its natural log becomes more negative There is no limit to how large negatively the log can get When the shrinking positive number actually reaches 0, its natural log is no longer defined in the set of real numbers (Perhaps it s non-real but complex, or maybe it s some other kind of number entirely Evaluating it is beyond the scope of this book)
Question 29-7
How can logarithms be used to change products into sums, or ratios into differences Do these properties of logs depend on the base
Answer 29-7
The logarithm of the product of two numbers is equal to the sum of their logarithms The logarithm of the ratio of two numbers is equal to the difference between their logarithms These rules work for common logs as well as for natural logs In fact, they work no matter what the base happens to be, as long as we don t change the base during the calculation!
Question 29-8
What is the common exponential of a number What is the natural exponential of a number
542 Review Questions and Answers Answer 29-8
The common exponential of a number is what we get when we raise 10 to a power equal to that number The natural exponential of a number is what we get when we raise e to a power equal to that number When working with natural exponentials, it s customary to call e the exponential constant
Question 29-9
How can we find the number x whose common exponential is 100,000 How can we find the number y whose natural exponential is 100,000
Answer 29-9
To find the number x whose common exponential is 100,000, we must find the power of 10 that gives us 100,000 We want to solve the equation 10 x = 100,000 It s easy see that x = 5 in this case But if we want to go through the motions of solving the above equation formally, we can take the common log (symbolized log10) of both sides, obtaining log10(10 x) = log10 100,000 The common log function undoes the common exponential function, so we can simplify this equation to x = log10 100,000 A calculator tells us that log10 100,000 is exactly equal to 5 Finding the number y whose natural exponential is 100,000 is a little more involved We want to find the power of e that gives us 100,000, so we must solve the equation e y = 100,000 If we take the natural log (symbolized ln) of both sides of this equation, we get ln (e y) = ln 100,000 The natural log function undoes the natural exponential function, so we have y = ln 100,000 A calculator tells us that y = 11513, rounded off to three decimal places
Question 29-10
How can we find the number x whose natural exponential is 1/e 5 How can we find the number y whose common exponential is 1/e 5
Part Three 543 Answer 29-10
To find the number x whose natural exponential is 1/e 5, we must find the power of e that gives us 1/e 5 We want to solve the equation e x = 1/e 5 This is almost trivial, because 1/e 5 is just another way of writing e 5 Now we have e x = e 5 Obviously, this means x = 5 If, despite the simplicity of this, we insist on solving formally and including every step, we can take the natural log of both sides of the above equation, obtaining ln (e x) = ln (e 5) We can simplify both sides to get x = 5 Finding the number y whose common exponential is 1/e 5 requires more work, but not much We want to find the power of 10 that gives us e 5, so we must solve the equation 10y = e 5 If we take the common log of both sides of this equation, we obtain log10 (10 y) = log10 (e 5) The common log function undoes the common exponential function, so we have y = log10 (e 5) A calculator tells us that e 5 = 0006737947, rounded off to nine decimal places That ought to be plenty of digits to give us a good idea of the final answer, which is the common log of 0006737947 Rounding off the end result to three decimal places, we get y = log10 (0006737947) = 2171
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