# Common Logarithm

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

## Definition

The common logarithm is a logarithm

with the base 10.

In high school math,

the base 10 is omitted.

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

## Example 1: log 5 = ? (log 2 = 0.301)

To make the factor 10,

change 5 to 10/2.

log 10/2 = log 10 - log 2

Logarithm of a quotient

log 10 = log_{10} 10

So log 10 = 1.

Logarithm of the base

log 2 = 0.301

1 - 0.301 = 0.699

So log 5 = 0.699.

This means 5 = 10^{0.669}.

Logarithmic form

## Example 2: log 120 = ? (log 2 = 0.301, log 3 = 0.477)

To make the factors 2, 3, and 10,

change 120 to 2^{2}⋅3⋅10

You don't need to split 10 to 2⋅5.

It'll be easier to use the factor 10.

log 2^{2}⋅3⋅10 = log 2^{2} + log 3 + log 10

Logarithm of a product

log 2^{2} = 2 log 2

Logarithm of a power

log 10 = 1

log 2 = 0.301

log 3 = 0.477

Then (given) = 2⋅0.301 + 0.477 + 1.

2⋅0.301 = 0.602

0.477 + 1 = 1.477

0.602 + 1.477 = 2.079

So log 120 = 2.079.

This means 120 = 10^{2.079}.

## Example 3: 2^{30} in Scientific Notation

Start from common loging the given number:

log 2^{30}.

Then log 2^{30} = 30 log 2.

Logarithm of a power

log 2 = 0.301

So (right side) = 30⋅0.301

30⋅0.301 = 9.03

Split 9.03 to 9 and 0.03.

log 2^{30} = 9 + 0.03

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

Logarithmic form

10^{9 + 0.03} = 10^{9} × 10^{0.03}

Product of powers

It says log 1.07 = 0.03.

Then 1.07 = 10^{0.03}.

So 10^{0.03} = 1.07.

So 2^{30} = 1.07 × 10^{9}.

As you can see,

by using common logarithm,

you can write the given number

in scientific notation.

Scientific notation

## Example 4: 3^{-20} in Scientific Notation

Start from common loging the given number:

log 3^{-20}.

Then log 3^{-20} = -20 log 3.

log 3 = 0.477

So (right side) = -20⋅0.477

20⋅0.477 = 9.54

Then (right side) = -9.54.

Split -9.54 to -9 and -0.54.

-0.54 part should be between 1 and 10.

(It becomes the [*a*] part of scientific notation.)

Change Number to Scientific notation

So write -9,

write -1 behind -9,

to undo the -1, write +1,

then write -0.54.

So (given) = -9 - 1 + 1 - 0.54.

-9 - 1 = -10

+1 - 0.54 = +0.46

So (given) = -10 + 0.46.

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

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

10^{-10 + 0.46} = 10^{-10} × 10^{0.46}

It says log 2.88 = 0.46.

Then 2.88 = 10^{0.46}.

So 10^{0.46} = 2.88.

So 3^{-20} = 2.88 × 10^{-10}.