# 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,

[log_{10}] is written as [log].

## Example 1

### Example

### Solution

5 = 10/2

log 10/2 = log 10 - log 2

Logarithm of a Quotient

log 10

= log_{10} 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

So 0.699 is the answer.

This means

10^{0.699} = 5.

## Example 2

### Example

### Solution

120 = 12⋅10

Write the prime factorization of 12.

12 = 2^{2}⋅3

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

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

= +log_{10} 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

So 2.079 is the answer.

This means

10^{2.079} = 120.

## Example 3

### Example

### Solution

First, common log 2^{30}.

log 2^{30} = 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 2^{30} = 9 + 0.03

The exponent is 9 + 0.03.

This is a common log.

So the base is 10.

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

Logarithmic Form

10^{9 + 0.03} = 10^{9} × 10^{0.03}

Product of Powers

To simplify 10^{0.03},

see log 1.07 = 0.03.

The exponent is 0.03.

The base is 10.

So 1.07 = 10^{0.03}.

1.07 = 10^{0.03}

So ×10^{0.03} = ×1.07.

Switch 10^{9} and 1.07.

So 1.07 × 10^{9} is the answer.

## Example 4

### Example

### 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} × 10^{0.46}

To simplify 10^{0.46},

see log 2.88 = 0.46.

The exponent is 0.46.

The base is 10.

So 2.88 = 10^{0.46}.

2.88 = 10^{0.46}

So ×10^{0.46} = ×2.88.

Switch 10^{-10} and 2.88.

So 2.88 × 10^{-10} is the answer.