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What Are Rational Inequalities?
What Is a Logarithm?
What Is Euler's Number e?
What Is the Natural Logarithm?
What Are the Logarithm Rules?

As you can see below, all the logarithmic rules for the common logarithm log ⁡ x and the natural logarithm ln ⁡ x are the same:

Common logarithm:

10log ⁡ a = a

Natural logarithm:

eln ⁡ a = a

This is true for any logarithm regardless of its base.

Rule

The Logarithmic Rules for the Common Logarithm

The first logarithmic rule:

log ⁡ (ax) = x log ⁡ a

The second logarithmic rule:

log ⁡ (a ⋅ b) = log ⁡ a + log ⁡ b

The third logarithmic rule:

log ⁡ (a b) = log ⁡ a − log ⁡ b

Rule

The Logarithmic Rules for the Natural Logarithm

The first logarithmic rule

ln ⁡ (ax) = x ln ⁡ a

The second logarithmic rule

ln ⁡ (a ⋅ b) = ln ⁡ a + ln ⁡ b

The third logarithmic rule

ln ⁡ (a b) = ln ⁡ a − ln ⁡ b

Example 1

Simplify log ⁡ a2 + log ⁡ b2 − 2 log ⁡ a

log ⁡ a2 + log ⁡ b2 −2 log ⁡ a = 2 log ⁡ a + 2 log ⁡ b −2 log ⁡ a = 2 log ⁡ b

log ⁡ a2 + log ⁡ b2 − 2 log ⁡ a = 2 log ⁡ a + 2 log ⁡ b − 2 log ⁡ a = 2 log ⁡ b

Example 2

Simplify log ⁡ ab + log ⁡ b2 − log ⁡ a2b

log ⁡ ab + log ⁡ b2 − log ⁡ a2b = log ⁡ a + log ⁡ b + 2 log ⁡ b −(log ⁡ a2 + log ⁡ b) = log ⁡ a + 3 log ⁡ b − log ⁡ a2 − log ⁡ b = log ⁡ a + 2 log ⁡ b − 2 log ⁡ a = − log ⁡ a + 2 log ⁡ b

log ⁡ ab + log ⁡ b2 − log ⁡ a2b = log ⁡ a + log ⁡ b + 2 log ⁡ b −(log ⁡ a2 + log ⁡ b) = log ⁡ a + 3 log ⁡ b − log ⁡ a2 − log ⁡ b = log ⁡ a + 2 log ⁡ b − 2 log ⁡ a = − log ⁡ a + 2 log ⁡ b

Example 3

Simplify log ⁡ a b − log ⁡ 2a b3

log ⁡ a b − log ⁡ 2a b3 = log ⁡ a − log ⁡ b −(log ⁡ 2a − log ⁡ b3) = log ⁡ a − log ⁡ b − (log ⁡ 2 + log ⁡ a − 3 log ⁡ b) = log ⁡ a − log ⁡ b − log ⁡ 2 − log ⁡ a + 3 log ⁡ b = 2 log ⁡ b − log ⁡ 2

log ⁡ a b − log ⁡ 2a b3 = log ⁡ a − log ⁡ b −(log ⁡ 2a − log ⁡ b3) = log ⁡ a − log ⁡ b − (log ⁡ 2 + log ⁡ a − 3 log ⁡ b) = log ⁡ a − log ⁡ b − log ⁡ 2 − log ⁡ a + 3 log ⁡ b = 2 log ⁡ b − log ⁡ 2

Example 4

Simplify log ⁡ 2x + log ⁡ 2 − log ⁡ 2 x2 + log ⁡ 10

log ⁡ 2x + log ⁡ 2 − log ⁡ 2 x2 + log ⁡ 10 = log ⁡ 2 + log ⁡ x + log ⁡ 2 − (log ⁡ 2 − log ⁡ x2) + 1 = 2 log ⁡ 2 + log ⁡ x − log ⁡ 2 + log ⁡ x2 + 1 = log ⁡ 2 + log ⁡ x + 2 log ⁡ x + 1 = log ⁡ 2 + 3 log ⁡ x + 1

log ⁡ 2x + log ⁡ 2 − log ⁡ 2 x2 + log ⁡ 10 = log ⁡ 2 + log ⁡ x + log ⁡ 2 −(log ⁡ 2 − log ⁡ x2) + 1 = 2 log ⁡ 2 + log ⁡ x − log ⁡ 2 + log ⁡ x2 + 1 = log ⁡ 2 + log ⁡ x + 2 log ⁡ x + 1 = log ⁡ 2 + 3 log ⁡ x + 1

Example 5

Use the logarithmic rules to simplify ln ⁡ 2x − ln ⁡ (x 2) − 4 ln ⁡ x

= ln ⁡ 2x − ln ⁡ (x 2 ) − 4 ln ⁡ x = ln ⁡ 2 + ln ⁡ x −(ln ⁡ x − ln ⁡ 2) − 4 ln ⁡ x = ln ⁡ 2 + ln ⁡ x − ln ⁡ x + ln ⁡ 2 − 4 ln ⁡ x = 2 ln ⁡ 2 − 4 ln ⁡ x

ln ⁡ 2x − ln ⁡ (x 2 ) − 4 ln ⁡ x = ln ⁡ 2 + ln ⁡ x −(ln ⁡ x − ln ⁡ 2) − 4 ln ⁡ x = ln ⁡ 2 + ln ⁡ x − ln ⁡ x + ln ⁡ 2 − 4 ln ⁡ x = 2 ln ⁡ 2 − 4 ln ⁡ x

Example 6

Use the logarithmic rules to simplify ln ⁡ 2x3 − ln ⁡ (3x 2 ) + ln ⁡ (3x)2

= ln ⁡ 2x3 − ln ⁡ (3x 2 ) + ln ⁡ (3x)2 = ln ⁡ 2 + ln ⁡ x3 −(ln ⁡ 3x − ln ⁡ 2) + ln ⁡ 32x2 = ln ⁡ 2 + 3 ln ⁡ x −(ln ⁡ 3 + ln ⁡ x − ln ⁡ 2) + ln ⁡ 32 + ln ⁡ x2 = ln ⁡ 2 + 3 ln ⁡ x − ln ⁡ 3 − ln ⁡ x + ln ⁡ 2 + 2 ln ⁡ 3 + 2 ln ⁡ x = 2 ln ⁡ 2 + 4 ln ⁡ x + ln ⁡ 3

ln ⁡ 2x3 − ln ⁡ (3x 2 ) + ln ⁡ (3x)2 = ln ⁡ 2 + ln ⁡ x3 −(ln ⁡ 3x − ln ⁡ 2) + ln ⁡ 32x2 = ln ⁡ 2 + 3 ln ⁡ x −(ln ⁡ 3 + ln ⁡ x − ln ⁡ 2) + ln ⁡ 32 + ln ⁡ x2 = ln ⁡ 2 + 3 ln ⁡ x − ln ⁡ 3 − ln ⁡ x + ln ⁡ 2 + 2 ln ⁡ 3 + 2 ln ⁡ x = 2 ln ⁡ 2 + 4 ln ⁡ x + ln ⁡ 3

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What Are Exponential Equations?
What Are Logarithmic Equations?
How to Solve Exponential Inequalities
How to Solve Logarithmic Inequalities