Part Three 541Answer 29-5
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.
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.
Question 29-6
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 actu-
ally becomes 0?
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?