104 LearnersLast updated on July 15th, 2025
Properties of Log
Logarithms are a way to simplify complex calculations involving exponents. The properties of logarithms help us rewrite and solve equations with exponents easily. These properties help us break large logarithms into smaller parts. Combine smaller logarithms into one and easily handle multiplication, division, and powers. We can apply these properties to calculate compound interest, measure sound levels, and understand growth patterns. In this topic, we will discuss more about the properties of Log.
What are the Properties of Log?
The properties of logs are simply rules that help us understand and work with logs. These rules are derived from the laws of exponents. There are 5 important rules we use to solve logarithmic equations.
Let’s take a look at these properties for better understanding:
1. Product Property: The log of two numbers multiplied together is the sum of their logs.
loga (mn) =loga m + loga n
2. Quotient Rule: The log of a number divided by another number is the difference of their logs.
loga (m/n) =loga m - loga n
3. Power Rule: The log of a number raised to a power is the power times the log of the base.
loga (mn) =n loga m
4. Change of base formula: We can change the base of the log using a different base, like 10 or e.
logba = logc a / logc b
5. Reciprocal Property: This property states that the logarithm of the reciprocal of a number is the negative of the log of the number itself.
loga (1m) = -loga (m)
There are additional properties of logarithms that are derived from exponent rules directly. Apart from the ones that are mentioned above.
- a0= 1 = loga 1 = 0
- a1= a = loga a = 1
- aloga x= x
- logbn am = m/n logba
Tips and Tricks for Properties of Log
Logarithms help us solve problems related to large numbers and exponents by making calculations easier. Understanding a few simple tips and tricks can help kids solve logarithmic problems faster and better. Let’s explore some ways to remember and use the properties of logs.
Log of 1 is always 0: No matter the base, the log of 1 is always 0. For example, log7 1 = 0.
Break Multiplication into Addition: When multiplying numbers inside a log, break them into a sum of logs. For example, Log2 (8 × 4) = log2 8 + log2 4
Power becomes a multiplier: If a number inside a log has an exponent, bring the exponent to the front. For example, log4 (16)2 = 2 log4 16.
Reciprocal Rule: The log of a fraction is the negative log of the whole number. For example, log5 12/5= - log5 25
Common Mistakes and How to Avoid Them in Properties of Log
It is common to make mistakes while learning logarithms. Let’s take a look at the common mistakes and how to avoid them to get a better understanding of the properties of logs.
Mistake 1
Forgetting the base
Sometimes students tend to forget to write the base. They write log 100 without mentioning the base, thinking it’s always base 10.
For example, instead of writing log10100 = 2, they write log2 100 ≠ 2.
Mistake 2
Incorrectly applying the product rule
Children improperly apply the product rule, thinking log a (mn) is the same as loga m. loga n.
For example, log2 (4 × 8) = log2 4 + log2 8, not their multiplication.
Mistake 3
Confusing the reciprocal property
Students assume that loga(1/m)= 1/loga m, but instead they should apply the rule correctly, which is loga(1/m)= - logam.
For example, instead of writing log2 1/8 = — log2 8 they write it as log2 (1/8) = 1/log28 = 1/3
Mistake 4
Incorrect expansion of logarithms
Students sometimes assume that log (a + b) = log a + logb which is incorrect. Logarithms do not distribute over addition. The correct rule is log (ab) = log a + log b.
Mistake 5
Assuming log x =x
Children assume that log x is equal to x. Understand that the logarithm is an operation and not just a variable.
Properties of Log Examples
Problem 1
Simplify log2 (8 × 4).
After simplifying the equation log2 (8 × 4), we get 5
Explanation
Using the product property of logarithms:
loga (mn) =loga m +loga n
loga (8×4) =log2 8 +log2 4
Since, 23 = 8 and 22 = 4, we get:
3 + 2 =5.
Problem 2
Simplify log₅ (25/5)
Simplifying log5 (25/5) we get, 1.
Explanation
Use the quotient property:
loga (mn) =loga m -loga n
loga (25/5) =log5 25 -log5 5
Since, 52 = 25 and 51 = 5, we get:
2 – 1 = 1.
Problem 3
Simplify log₃ (27²)
Simplifying log3 (272) we get 6.
Explanation
Using the power property:
loga (mn) =n loga m
log3 (272) =2 log3 27
Since 33 = 27, we get:
2 × 3 = 6
Problem 4
Evaluate log₇ 1.
The value of log7 1 we get 0.
Explanation
By the logarithm rule:
logb 1 = 0
log7 1 = 0
Problem 5
Simplify log₃ (9 × 27)
Simplifying log3(9 × 27), we get 5.
Explanation
Apply the product rule,
log3(9 × 27) = log3 9 + log3 27
Log3 9 =2, log3 27=3
2 + 3 = 5
FAQs on Properties of Log
1.What is the logarithm of zero?
2.What is the logarithm of the base itself?
3.Can logarithms be decimal or fractions?
4.Can logarithms have decimal bases?
5.What does logb 1 always equal?
6.How does learning Algebra help students in Australia make better decisions in daily life?
7.How can cultural or local activities in Australia support learning Algebra topics such as Properties of Log?
8.How do technology and digital tools in Australia support learning Algebra and Properties of Log?
9.Does learning Algebra support future career opportunities for students in Australia?
Important Glossaries for Properties of Log
- Logarithm : It is a mathematical operation that determines the exponent needed to raise a given base to obtain a specific number. For example, log2 8 = 3 (since 23 = 8).
- Base: The number that is raised to a power in exponential form. For example, In log327, the base is 3.
- Exponent: The power to which a number (base) is raised. For example, In 23= 8, the exponent is 3.
Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
