

What Are the 7 Logarithm Rules and How Do They Work?
The concept of log rules is essential in mathematics and helps in solving real-world and exam-level problems efficiently. Mastering log rules allows students to simplify complex logarithmic expressions, making calculations easier both in classwork and competitive exams.
Understanding Log Rules
A log rule refers to one of several fundamental properties that govern how logarithms can be simplified, expanded, or combined. These rules are widely used in algebra, calculus, and competitive exam preparation. Some important related concepts include properties of logarithms, natural logs (ln), and log derivative rules. With solid understanding, students can solve complicated log problems quickly and accurately.
Log Rules Table
Here’s a helpful table to understand log rules more clearly:
Key Logarithm Rules
Rule Name | Formula | Explanation (in Words) |
---|---|---|
Product Rule | logb(mn) = logbm + logbn | Log of a product equals sum of logs |
Quotient Rule | logb(m/n) = logbm - logbn | Log of a quotient equals difference of logs |
Power Rule | logb(mn) = n logbm | Log of a power brings exponent in front |
Change of Base | logab = logcb / logca | Convert logs to another base |
Log of 1 | logb1 = 0 | Log of 1 is always zero |
Log of the Base | logbb = 1 | Log of base to itself is one |
Exponential Rule | blogbx = x | Exponent and log cancel each other |
This table shows the seven important log rules that appear regularly in board exams and competitive tests.
Derivation and Explanation of Log Rules
Let’s see how the basic log rules are derived using exponent laws:
1. Product Rule:
So, m = bx and n = by.
mn = bx × by = bx+y.
Now, logb(mn) = x + y = logbm + logbn.
2. Quotient Rule:
So, m = bx and n = by.
m/n = bx / by = bx-y.
Now, logb(m/n) = x - y = logbm - logbn.
3. Power Rule:
So, m = bx.
mn = (bx)n = bnx.
So, logb(mn) = n x = n logbm.
Natural Log Rules (ln)
Natural logarithm uses base "e" and is written as ln. The same log rules apply for natural logs:
ln Rule | Formula |
---|---|
Product | ln(mn) = ln m + ln n |
Quotient | ln(m/n) = ln m - ln n |
Power | ln(mn) = n ln m |
ln(e) | ln e = 1 |
ln(1) | ln 1 = 0 |
For more on natural logs, visit Difference Between Log and ln.
Worked Examples of Log Rules
See how to use log rules step by step:
Example 1: Simplify: log236 + log25
2. Multiply: 36 × 5 = 180
Final Answer: log2180
Example 2: Express ln 72 in terms of p and q if p = ln 2, q = ln 6
2. ln 72 = ln(62 × 2) = ln 62 + ln 2
3. ln 62 = 2 ln 6
4. So, ln 72 = 2 ln 6 + ln 2 = 2q + p
Final Answer: 2q + p
Practice Problems
- Simplify: log52 + log58
- Compress: 2 log3x − log3y
- If log10x = 2, find x
- Evaluate: ln(e5)
- Express log216 using log rules
For more questions, try this log worksheet for practice.
Common Mistakes to Avoid
- Trying to split log(m + n). There is no log rule for addition: log(m + n) ≠ log m + log n!
- Confusing log and ln. ln uses base e, log usually means base 10.
- Forgetting to check the log base before applying rules.
- Missing sign when applying the quotient rule.
To learn more about log properties and avoid mistakes, see Properties of Logarithms.
Log Rules and Exponents
Logarithms and exponents are closely related. For example, logb(mn) = n logbm links exponent rules with log rules. To deepen your understanding, check out Exponents and Laws of Exponents.
Log Rules in Calculus
Log rules are also used in calculus, especially for differentiation. For instance, if y = ln x, then dy/dx = 1/x. To see detailed steps, read about logarithmic differentiation.
Real-World Applications
Log rules help in calculations for earthquakes (Richter scale), sound intensity, population growth, banking (compound interest), and more. Vedantu shows how mathematics applies beyond just exams.
We explored the idea of log rules, how to apply them, stepwise solutions, and their real-life usage. Practice regularly with Vedantu for clear concepts and exam success!
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FAQs on The 7 Rules and Properties of Logarithms for Students
1. What are the 7 rules of logarithms?
The 7 rules of logarithms are fundamental properties used to simplify and solve logarithmic expressions:
1. Product Rule: logb(A × B) = logbA + logbB
2. Quotient Rule: logb(A/B) = logbA − logbB
3. Power Rule: logb(An) = n × logbA
4. Change of Base Rule: logaB = logcB / logcA
5. Log of 1: logb1 = 0
6. Log of base: logbb = 1
7. Inverse Rule: If logba = x, then bx = a. These rules are important for simplifying logarithmic expressions, solving equations, and calculus applications.
2. What are the 5 log rules?
The 5 essential log rules commonly used are:
1. Product Rule: logb(A × B) = logbA + logbB
2. Quotient Rule: logb(A/B) = logbA − logbB
3. Power Rule: logb(An) = n × logbA
4. Log of 1 is zero: logb1 = 0
5. Log of base is one: logbb = 1. These are the most frequently used properties in solving logarithmic equations for CBSE, ICSE, and competitive exams like Math Olympiad, JEE, and MCAT.
3. What are the 7 properties of logarithms?
The 7 properties of logarithms are the same as the 7 logarithm rules and include:
1. Product Rule
2. Quotient Rule
3. Power Rule
4. Change of Base Rule
5. Log of 1
6. Log of base
7. Inverse Rule. These properties help simplify, expand, or compress logarithmic expressions in mathematics and calculus.
4. What is the base 10 rule for logarithms?
The base 10 logarithm, also called the common logarithm and denoted as log(x), has special rules:
- log1010 = 1
- log101 = 0
Base 10 logs are widely used in science and engineering for representing large values with powers of 10. All general log rules also apply to log base 10.
5. What is the product rule of logarithms?
The Product Rule for logarithms states: logb(A × B) = logbA + logbB. It allows you to separate multiplication inside a logarithm into an addition of separate logs at the same base.
6. What is the quotient rule for logarithms?
The Quotient Rule for logarithms is: logb(A/B) = logbA − logbB. This property means the division inside the log turns into subtraction of two logs with the same base.
7. What is the power rule of logarithms?
The Power Rule for logarithms states: logb(An) = n × logbA. You can bring any exponent in the argument down as a multiplier in front of the log.
8. What are the natural logarithm (ln) rules?
The natural logarithm rules (for ln, base e) are similar to other log rules:
- ln(AB) = ln A + ln B
- ln(A/B) = ln A − ln B
- ln(An) = n × ln A
- ln(1) = 0
- ln(e) = 1. Natural logs are important in advanced mathematics, calculus, and science because e is the base of natural growth and decay.
9. How do you differentiate logarithmic functions (log rules calculus)?
The derivative of a logarithmic function is:
- For natural log: d/dx [ln x] = 1/x
- For base b: d/dx [logb x] = 1/(x ln b)
Use the chain rule and properties of log derivatives to solve calculus questions involving logs.
10. What is the log(a+b) rule?
There is no direct rule for log(a + b). The logarithm properties only apply to products, quotients, and powers:
log(a + b) ≠ log a + log b. To simplify log(a + b), you must use special algebraic methods or estimation, not log rules.
11. What is the change of base rule in logarithms?
The Change of Base Rule in logarithms allows switching between bases:
logab = logcb / logca
You can calculate any logarithm with a calculator using this rule, as most calculators support only log base 10 and ln (base e).
12. How do you use log rules to solve equations?
To solve log equations:
1. Use properties (product, quotient, power) to simplify
2. Combine all logs into a single log, if possible
3. Rewrite in exponential form to solve for the variable
4. Check solutions by substituting back, as log arguments must be positive.
These steps will help you correctly solve logarithmic equations for CBSE, engineering, and entrance exams.

















