# Common Logarithm

How to use the common logarithm to find the value of a number and write a number in scientific notation: formula, 4 examples, and their solutions.

## Definition

### Definition

A common logarithm (common log)
is a logarithm
whose base is 10.

In high school math,
[log10] is written as [log].

## Example 1

### Solution

5 = 10/2

log 10/2 = log 10 - log 2

Logarithm of a Quotient

log 10
= log10 10
= 1

Logarithm of Itself

It says
assume log 2 = 0.301.

So -log 2 = -0.301.

So
log 10 - log 2
= 1 - 0.301.

1 - 0.301 = 0.699

This means
100.699 = 5.

## Example 2

### Solution

120 = 12⋅10

Write the prime factorization of 12.

12 = 22⋅3

log (22⋅3⋅10)
= log 22 + log 3 + log 10.

Logarithm of a Product

log 22 = 2 log 2

Logarithm of a Power

It says
assume log 2 = 0.301.

So 2 log 2 = 2⋅0.301.

It says
assume log 3 = 0.477.

So +log 3 = +0.477.

+log 10
= +log10 10
= 1

So
2 log 2 + log 3 + log 10
= 2⋅0.301 + 0.477 + 1.

2⋅0.301 = 0.602

0.602 + 0.477 = 1.079

1.079 + 1 = 2.079

This means
102.079 = 120.

## Example 3

### Solution

First, common log 230.

log 230 = 30 log 2

It says
assume log 2 = 0.301.

So 30 log 2 = 30⋅0.301.

30⋅0.301 = 9.03

Split 9.03 into
the integer 9
and +0.03.

This +0.03 should be between 0 and 1.

log 230 = 9 + 0.03

The exponent is 9 + 0.03.

This is a common log.
So the base is 10.

Then 230 = 109 + 0.03.

Logarithmic Form

109 + 0.03 = 109 × 100.03

Product of Powers

To simplify 100.03,
see log 1.07 = 0.03.

The exponent is 0.03.
The base is 10.

So 1.07 = 100.03.

1.07 = 100.03

So ×100.03 = ×1.07.

Switch 109 and 1.07.

So 1.07 × 109 is the answer.

## Example 4

### Solution

First, common log 3-20.

log 3-20 = -20 log 3

It says
assume log 3 = 0.477.

So -20 log 3 = -20⋅0.477.

-20⋅0.477 = -9.54

Split -9.54 into
the integer -9
and -0.54.

-0.54 should be between 0 and 1.
But it's not between 0 and 1.

Then, write -1 and +1
between -9 and -0.54.

-9 - 1 = -10
+1 - 0.54 = +0.46

Now this +0.46 is between 0 and 1.

log 3-20 = -10 + 0.46

The exponent is -10 + 0.46.

This is a common log.
So the base is 10.

Then 3-20 = 10-10 + 0.46.

10-10 + 0.46 = 10-10 × 100.46

To simplify 100.46,
see log 2.88 = 0.46.

The exponent is 0.46.
The base is 10.

So 2.88 = 100.46.

2.88 = 100.46

So ×100.46 = ×2.88.

Switch 10-10 and 2.88.

So 2.88 × 10-10 is the answer.