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LOGARITHMS
A logarithm (or log, for short) is a number that represents a power or exponent. In this book, we will focus on only base 10 logs, also called common logs, which are defined as follows:
log10 x is the power to which 10 must be raised to obtain x.
You may find it easier to remember the meaning with a less technical definition:
log10 x means "10 to what power equals x?"
For example:
Four important rules follow directly from the definition of a logarithm.
1. Taking the logarithm of a power of 10 gives the power. That is,
log10 10x = x
2. Raising 10 to a power that is the logarithm of a number gives back the number. That is,
3. Because powers of 10 are multiplied by adding their exponents, we have the addition rule for logarithms:
4. We can “bring down” an exponent within a logarithm by applying the power rule for logarithms:
Most calculators have a key to compute log10 of any positive number. You should find this key on your calculator and use it to verify that log10 1000 = 3 and log102 = 0.301030.
Example: Given that log102 = 0.301030, find each of the following:
a. log10 8 b.  c. log10 200
Solution:
a. We notice that 8 = 23. Thus, from Rule 4,
b. From Rule 2,
c. We notice that  Thus, from Rule 3,
From Rule 1, we know that log10 102 = 2, so
Example: Someone tells you that log10 600=5.778. Should you believe it?
Solution: Because 600 is between 100 and 1000, log10 600 must be between log10 100 and log10 1000. From Rule 1, we find that log10 100 = log10 102 = 2 and log10 1000 = log10 103 = 3. Thus, log10 600 must be between 2 and 3, so the claimed answer of 5.778 must be wrong.
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