Algebra Know-It-ALL

540 Review Questions and Answers

• The common log of 1,000 is 3, because 10^3 = 1,000.


• The common log of 10,000 is 4, because 10^4 = 10,000.

As a number gets larger without limit, so does its common log. The size of the logarithm

grows much more slowly than the size of the number.

Question 29-3

According to the definition in Answer 29-1, What is the common log of 1? Of 1/10? Of 1/100?

Of 1/1000? What happens to the common log of a positive real number whose absolute value

keeps shrinking, that is, as the number approaches 0 from the positive direction? What hap-

pens to the common log of a shrinking positive real number when it actually becomes 0?

Answer 29-3

As the absolute value of a positive number keeps shrinking, its common log changes like this:

• The common log of 1 is 0, because 10^0 = 1.


• The common log of 1/10 is −1, because 10−^1 = 1/10.


• The common log of 1/100 is −2, because 10−^2 = 1/100.


• The common log of 1/1,000 is −3, because 10−^3 = 1/1,000.

As a shrinking positive number approaches 0, its common log becomes more negative.

As the shrinking positive number “closes in” on 0, the common log decreases—that is, it

increases negatively—without limit. When the shrinking positive number actually reaches

0, its common 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 29-4

If we say that the natural logarithm of a certain number p is equal to q, what do we mean?

Answer 29-4

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 vari-

ous powers. The value of e is approximately 2.71828. It’s an irrational number, however, so it

cannot be fully written out in decimal form.

Question 29-5

According to the definition in Answer 29-4, what is the natural log of e? Of e^2? Of e^3? Of e^4?

What happens to the natural log of a number as that number grows larger without limit?