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
Example
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
So 0.699 is the answer.
This means
100.699 = 5.
Example 2
Example
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
So 2.079 is the answer.
This means
102.079 = 120.
Example 3
Example
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
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 × 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.