Common Logarithms

Common Logarithms

How to use common logarithms to find the value of a number: definition, examples, and their solutions.

Definition

The common logarithm is a logarithm with base 10. log_10 c = log c.

The common logarithm is a logarithm
with base 10.

In high school math,
the base 10 is omitted.

So log10 c = log c.

Example 1

Find the value of log 5. (Assume log 2 = 0.301.)

To find the factor 10 from 5,
change 5 to 10/2.

Then log 10/2 = log 10 - log 2.

Logarithmic of quotients

log 10 = log10 10 = 1.

And log 2 = 0.301.
(given condition)

So log 10 - log 2 = 1 - 0.301.

log 5 = 0.669 means
5 = 100.669.

Example 2

Find the value of 2^30. (Assume log 2 = 0.301, log 1.07 = 0.03.)

Find the value of log 230.

Logarithms of powers

log 2 = 0.301
(given condition)

Split 9 and +0.03.

Solve the common log.

Then 230 = 109 + 0.03
= 109⋅100.03

Product of powers

log 1.07 = 0.03
(given condition)

So 1.07 = 100.03.

See the relationship between log 230 = 9 + 0.03
and 230 = 109⋅1.07.

By using the common log,
you can find the value of a number
in scientific notation.

9: how big the number is. (= 109)
0.03: how the number looks like. (100.03 = 1.07)

Standard notation to scientific notation

Example 3

Find the value of 3^-20. (Assume log 3 = 0.477, log 2.88 = 0.46.)

Find the value of log 3-20.

Logarithms of powers

log 3 = 0.477
(given condition)

-9.54 = -9 - 0.54

But -0.54 is (-).

So make the latter term (+):
-9 - 0.54 = -10 + (1 - 0.54)
= -10 + 0.46

This is to make 1 ≤ 100.46 < 10.

Standard notation to scientific notation

Solve the common log.

Then 3-20 = 10-10 + 0.46
= 10-10⋅100.46

Product of powers

log 2.88 = 0.46
(given condition)

So 2.88 = 100.46.

-10: how big the number is. (= 10-10)
0.46: how the number looks like. (100.46 = 2.88)

Standard notation to scientific notation