Logarithm Rules (aka Log Laws) Explained with Examples

What is Logarithm Rules (Definition)

To understand Logarithm Rules, you first need to know what is Logarithm or Log.

What is Log or Logarithm?

Logarithm is basically a power on any number that we use to get the other number defined in the Log.

Examples:

Question: What is the value of log₂ 32?
Answer: 5

Explanation: 2⁵ = 32

We need to get 32 by multiplying 2 with 2 and see how many times it goes. Sp 2 x 2 x 2 x 2 x 2 = 32. This will get us 2⁵

This is the reason the value of log₂ 32 = 5

Question: How about log₃ 81?
Answer: 4 (because 3⁴ )

Tricky Question: Get the value of Log 1000?

Answer: 3 (But how?)

Explanation: When there is no base value associated with Log, we assume it’s log₁₀

so log₁₀ 1000 = 10³

and the answer is 3.

What are the Rules of Logarithm (Log Laws Table)

Name of the Rule	Rule
Product Rule of Logarithm	loga xy = loga x + loga y
Power Rule of Logarithm	loga x y = y loga x
Quotient/Ratio Rule of Logarithm	loga x / y = loga x – loga y
Base Switch Rule of Logarithm	loga b = 1 / logb a
Base Change Rule of Logarithm	loga(x) = logb(x) / logb a
Derivative of Logarithm	fx = loga x ⇒ f ‘ (x) = 1 / ( x ln(a) )
Integral of Logarithm	∫ loga x dx = x ( loga x – 1 / ln(a) ) + b
Logarithm of 1	loga 1 = 0
Logarithm of 0	loga 0 is undefined
Logarithm of the Base	loga b = 1
Logarithms of Infinity	lim loga x = ∞, when x→∞

Download the Log Table in Image Format or PDF Format

1. Solved Examples for Product Rule of Logarithm

Rule:   log_a xy = log_a x + log_a y

Question: Solve this: log₂ 4*16 using Log Law.
The same question can also be written as log₂ 4 + log₂ 16

Answer: 

log₂ 4*16

=> log₂ 4 + log₂ 16

=> log₂ 2 ² + log₂ 2 ⁴

=> 2log₂ 2 + 4log₂ 2

=> 2 * 1 + 4 * 1

=> 2 + 4

=> 6

so  log₂ 4*16 = 6

2. Solved Example for Power Rule of Logarithm

Rule:   log_a x ^y = y log_a x

Question: Solve this log₃(3²⁷)

This question can also be written as 27log₃3

Note: This type of questions are usually asked as objective questions because it doesn’t have much to do. Usually it is combined with other rules in question that we have also done after cover other rules. Scroll down if you are in rush to see.

Answer:

log₃(3²⁷)

=> 27log₃3

=> 27 * 1

=> 27

Answer is 27.

3. Solved Example for Quotient/Ratio Rule of Logarithm

Rule: log_a x / y = log_a x – log_a y

Question: Solve log₄ 1024 – log₄ 16

Answer: 

log₄ 1024 – log₄ 16

=> log₄ (1024 / 16)

=> log₄ 64

=> log₄ 4³

=> 3log₄ 4

=>3 * 1

=> 3

S0, log₄ 1024 – log₄ 16 = 3

4. Solved Example for Base Switch Rule

Rule:  log_a b = 1 / log_b a

Question: 1 / log₂ 128

Answer: 

1 / log₂ 128

=> 1 / log₂ 2⁷

=> 1 / 7 * 1

=> 1 / 7

=> 0.1429

Now let’s mix it up. Mix different types of Log Laws in one example.

5. Solved Example (Mixed of different Rules)

Question: Solve this: log₃ 9 + log₃ 81 – log₅ 1250 + log₅ 2

Answer:

log₃ 9 + log₃ 81 + log₅ 1250 – log₅ 2

=> log₃ (9 * 81) + log₅ (1250 / 2)

=> log₃ 729 + log₅ 625

=> log₃ 3⁶ + log₅ 5⁴

=> 6log₃ 3 + 4log₅ 5

=> 6*1 + 4*1

=> 6+4

=> 10

So the value of log₃ 9 + log₃ 81 + log₅ 1250 – log₅ 2 = 10

6. Solved Exmple: Expand this log₈ (64_k 4 / _n9)

log₈ (64_k 4 / _n9)

=> log₈ (64_k 4) – log₈ n⁹

=> log₈ 64 + log₈ k⁴ – log₈ n⁹

=> log₈ 8⁸ + 4log₈ k – 9log₈ n

=> 8log₈ 8 + 4log₈ k – 9log₈ n

=> 8 * 1 + 4log₈ k – 9log₈ n

=> 8 + 4log₈ k – 9log₈ n

So, log₈ (64_k 4 / _n9) =  8 + 4log₈ k – 9log₈ n

Watch this video for examples and understanding of the Logarithm Law

Do you have other questions? You can type in the comment.