# 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.

In high school math,

the base 10 is omitted.

So log_{10} *c* = log *c*.

## Example 1

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 = log_{10} 10 = 1.

And log 2 = 0.301.

(given condition)

So log 10 - log 2 = 1 - 0.301.

log 5 = 0.669 means

5 = 10^{0.669}.

## Example 2

Find the value of log 2^{30}.

Logarithms of powers

log 2 = 0.301

(given condition)

Split 9 and +0.03.

Solve the common log.

Then 2^{30} = 10^{9 + 0.03}

= 10^{9}⋅10^{0.03}

Product of powers

log 1.07 = 0.03

(given condition)

So 1.07 = 10^{0.03}.

See the relationship between log 2^{30} = 9 + 0.03

and 2^{30} = 10^{9}⋅1.07.

By using the common log,

you can find the value of a number

in scientific notation.

9: how big the number is. (= 10^{9})

0.03: how the number looks like. (10^{0.03} = 1.07)

Standard notation to scientific notation

## Example 3

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 ≤ 10^{0.46} < 10.

Standard notation to scientific notation

Solve the common log.

Then 3^{-20} = 10^{-10 + 0.46}

= 10^{-10}⋅10^{0.46}

Product of powers

log 2.88 = 0.46

(given condition)

So 2.88 = 10^{0.46}.

-10: how big the number is. (= 10^{-10})

0.46: how the number looks like. (10^{0.46} = 2.88)

Standard notation to scientific notation