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# Common Logs

## Notation for Common Logs

A logarithm has the form logbx, where x and b are real numbers, x > 0, b > 0, and b 1.

A common logarithm has 10 for its base, b. Here are several examples of common logs:

 log 10 1000 log 10 43 log 10 0.87

Common logs are often abbreviated by not writing the base.

For example, the common logarithms shown above can be written as:

 log 1000 log 43 log 0.87

Definition â€” Common Logarithm

A common log is a log whose base is 10. It is written like this:

log x = log10 x Here, x > 0.

Common logs are useful since they have the same base as our number system, which is base 10.

## Finding Common Logs

We can find the value of some common logs by switching from logarithmic notation to exponential notation.

Example 1

Find: log 10,000

Solution

 The base of a common log is 10.Therefore, write log 10,000 as log10 10,000. Set x equal to log10 10,000. Rewrite in exponential form. Rewrite 10,000 as a power of 10. Use the Principle of Exponential Equality to solve for x. So, log 10,000 = 4. x = 10x = 10x = x = log10 10,000 log10 10,000 10,000 104 4

Note:

A factor tree can be used to write 10,000 as a power of 10.

Thus, 10,000 = 10 Â· 10 Â· 10 Â· 10 = 104.

Principle of Exponential Equality:

If bx = by, then x = y. Here, b > 0 and b 1.

In many cases we cannot find the value of a common log by switching from logarithmic notation to exponential notation.

For example, we can rewrite x = log 65 as 10x = 65, but it isnâ€™t easy to see what power of 10 equals 65.

We can approximate log 65 by noting the following:

1 = log10 10 is equivalent to 101 = 10

x = log10 65 is equivalent to 10x = 65

2 = log10 100 is equivalent to 102 = 100

So, log 65 must be between 1 and 2.

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