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 log₁₀ 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 = log₁₀ 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 2²⋅3⋅10

You don't need to split 10 to 2⋅5.
It'll be easier to use the factor 10.

log 2²⋅3⋅10 = log 2² + log 3 + log 10

Logarithm of a product

log 2² = 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: 2³⁰ in Scientific Notation

Start from common loging the given number:
log 2³⁰.

Then log 2³⁰ = 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 2³⁰ = 9 + 0.03

Then 2³⁰ = 10^{9 + 0.03}.

Logarithmic form

10^{9 + 0.03} = 10⁹ × 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 2³⁰ = 1.07 × 10⁹.

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.

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