The Logarithm Formula expresses the power to which a fixed base must be raised to produce a given number, written as \ ( \log_b x = y \) which means \ ( b^y = x \). Logarithm formulas are covered in NCERT Class 9 (Appendix) and thoroughly in Class 11 Mathematics (Chapter 1 — Sets, and supplementary material). For JEE Main and JEE Advanced aspirants, logarithm identities are indispensable tools for simplifying complex algebraic and exponential expressions. This article covers every key logarithm formula, a complete formula sheet, three progressive solved examples, CBSE exam tips 2025-26, common mistakes, and JEE/NEET applications.
Key Logarithm Formulas at a Glance
Quick reference for the most important logarithm formulas used in CBSE and competitive exams.
- Definition: \( \log_b x = y \iff b^y = x \)
- Product Rule: \( \log_b (mn) = \log_b m + \log_b n \)
- Quotient Rule: \( \log_b \left(\frac{m}{n}\right) = \log_b m – \log_b n \)
- Power Rule: \( \log_b (m^n) = n \log_b m \)
- Change of Base: \( \log_b m = \frac{\log_c m}{\log_c b} \)
- Natural Log: \( \ln x = \log_e x \)
- Common Log: \( \log x = \log_{10} x \)
What is the Logarithm Formula?
The Logarithm Formula defines the inverse relationship between exponentiation and logarithms. If a base \( b \) is raised to the power \( y \) to give \( x \), then the logarithm of \( x \) to the base \( b \) equals \( y \). In symbolic form:
\[ \log_b x = y \iff b^y = x \]
Here, \( b > 0 \), \( b \neq 1 \), and \( x > 0 \). Logarithms were introduced to simplify multiplication and division of large numbers into addition and subtraction. In the NCERT curriculum, logarithms first appear in Class 9 (Appendix 1) and are formally studied in Class 11 Mathematics. The two most commonly used bases are base 10 (common logarithm, written \( \log x \)) and base \( e \) (natural logarithm, written \( \ln x \)). Understanding the logarithm formula is essential for solving exponential equations, working with scientific notation, and analysing growth and decay problems in both CBSE board exams and competitive entrance tests like JEE and NEET.
Logarithm Formula — Expression and Variables
The fundamental logarithm formula and its key laws are listed below. Each law follows directly from the definition of a logarithm as an exponent.
\[ \log_b x = y \iff b^y = x \quad (b > 0,\ b \neq 1,\ x > 0) \]
| Symbol | Quantity / Meaning | Condition / Unit |
|---|---|---|
| \( b \) | Base of the logarithm | \( b > 0,\ b \neq 1 \) |
| \( x \) | Argument (the number whose log is taken) | \( x > 0 \) |
| \( y \) | Logarithm value (the exponent) | Any real number |
| \( \log \) | Common logarithm (base 10) | Base = 10 |
| \( \ln \) | Natural logarithm (base \( e \)) | Base = \( e \approx 2.718 \) |
Derivation of the Product Rule
Let \( \log_b m = p \) and \( \log_b n = q \). By definition, \( b^p = m \) and \( b^q = n \). Multiplying both sides gives \( mn = b^p \cdot b^q = b^{p+q} \). Converting back to logarithmic form: \( \log_b(mn) = p + q = \log_b m + \log_b n \). The quotient and power rules follow the same logic using division and repeated multiplication of exponents, respectively. This derivation shows that every logarithm law is a direct consequence of the laws of exponents taught in NCERT Class 8 and Class 9.
Complete Logarithm Formula Sheet
The table below is a comprehensive reference for all logarithm formulas tested in CBSE Class 9-12 and JEE/NEET examinations.
| Formula Name | Expression | Variables | Notes | NCERT Reference |
|---|---|---|---|---|
| Definition | \( \log_b x = y \iff b^y = x \) | b = base, x = argument, y = log value | b > 0, b ≠ 1, x > 0 | Class 11, Appendix |
| Product Rule | \( \log_b(mn) = \log_b m + \log_b n \) | m, n > 0 | Converts multiplication to addition | Class 11, Ch 1 |
| Quotient Rule | \( \log_b\!\left(\frac{m}{n}\right) = \log_b m – \log_b n \) | m, n > 0 | Converts division to subtraction | Class 11, Ch 1 |
| Power Rule | \( \log_b(m^n) = n\,\log_b m \) | n = any real number | Brings exponent down as coefficient | Class 11, Ch 1 |
| Change of Base Rule | \( \log_b m = \dfrac{\log_c m}{\log_c b} \) | c = any valid new base | Useful for calculator computation | Class 11, Ch 1 |
| Log of 1 | \( \log_b 1 = 0 \) | Any valid base b | Because \( b^0 = 1 \) | Class 9, Appendix 1 |
| Log of Base | \( \log_b b = 1 \) | Any valid base b | Because \( b^1 = b \) | Class 9, Appendix 1 |
| Reciprocal Rule | \( \log_b m = \dfrac{1}{\log_m b} \) | m, b > 0, m ≠ 1 | Special case of change of base | Class 11, Ch 1 |
| Exponential-Log Identity | \( b^{\log_b x} = x \) | x > 0 | Log and exponential are inverses | Class 11, Ch 1 |
| Natural Log Definition | \( \ln x = \log_e x \) | e ≈ 2.71828 | Used in calculus and NEET Biology | Class 12, Ch 5 |
| Log Base Conversion (10 to e) | \( \ln x = 2.303\,\log_{10} x \) | x > 0 | Frequently used in Chemistry (pH, rate laws) | Class 12 Chemistry, Ch 4 |
| Negative Argument of Log | \( \log_b\!\left(\frac{1}{x}\right) = -\log_b x \) | x > 0 | Derived from quotient rule with m = 1 | Class 11, Ch 1 |
Logarithm Formula — Solved Examples
Example 1 (Class 9-10 Level)
Problem: Evaluate \( \log_2 64 \).
Given: Base \( b = 2 \), Argument \( x = 64 \)
Step 1: Write 64 as a power of 2. We know \( 2^6 = 64 \).
Step 2: Apply the definition \( \log_b x = y \iff b^y = x \). Since \( 2^6 = 64 \), we get \( \log_2 64 = 6 \).
Answer
\( \log_2 64 = 6 \)
Example 2 (Class 11-12 Level)
Problem: Simplify \( \log_{10} 200 + \log_{10} 5 – \log_{10} 10 \).
Given: Expression involves three common logarithm terms.
Step 1: Apply the Product Rule to the first two terms.
\[ \log_{10} 200 + \log_{10} 5 = \log_{10}(200 \times 5) = \log_{10} 1000 \]
Step 2: Apply the Quotient Rule.
\[ \log_{10} 1000 – \log_{10} 10 = \log_{10}\!\left(\frac{1000}{10}\right) = \log_{10} 100 \]
Step 3: Evaluate \( \log_{10} 100 \). Since \( 10^2 = 100 \):
\[ \log_{10} 100 = 2 \]
Answer
\( \log_{10} 200 + \log_{10} 5 – \log_{10} 10 = 2 \)
Example 3 (JEE Level)
Problem: If \( \log_3 2 = a \) and \( \log_3 5 = b \), express \( \log_3 90 \) in terms of \( a \) and \( b \).
Given: \( \log_3 2 = a \), \( \log_3 5 = b \)
Step 1: Factorise 90 into prime factors. \( 90 = 2 \times 3^2 \times 5 \).
Step 2: Apply the Product Rule.
\[ \log_3 90 = \log_3 2 + \log_3 3^2 + \log_3 5 \]
Step 3: Apply the Power Rule to the middle term.
\[ \log_3 3^2 = 2\,\log_3 3 = 2 \times 1 = 2 \]
Step 4: Substitute the given values.
\[ \log_3 90 = a + 2 + b \]
Answer
\( \log_3 90 = a + b + 2 \)
CBSE Exam Tips 2025-26
- Memorise the seven core laws first. The product, quotient, power, change-of-base, reciprocal, log-of-1, and log-of-base rules cover nearly every CBSE question. We recommend writing them on a single revision card.
- Always check the domain. CBSE examiners deduct marks when students apply \( \log_b x \) without verifying \( x > 0 \) and \( b > 0, b \neq 1 \). State the condition in your answer.
- Convert to a single logarithm before evaluating. In 3-4 mark questions, combine all log terms using product and quotient rules before computing the final value. This reduces arithmetic errors.
- Use \( \log_{10} 2 \approx 0.3010 \) and \( \log_{10} 3 \approx 0.4771 \). CBSE Class 11 questions frequently ask you to evaluate expressions using these standard values. Memorise them for 2025-26 exams.
- Practice change-of-base conversions. Questions that mix bases (e.g., \( \log_2 \) and \( \log_8 \)) appear regularly. Convert all terms to a common base using the change-of-base formula before simplifying.
- Distinguish \( \log \) from \( \ln \) in application problems. In Chemistry and Biology-linked maths problems, \( \ln \) is used. Remember \( \ln x = 2.303 \log_{10} x \) for quick conversion.
Common Mistakes to Avoid
- Mistake 1 — Adding arguments instead of multiplying: Students often write \( \log_b(m + n) = \log_b m + \log_b n \). This is incorrect. The product rule applies to \( \log_b(mn) \), not \( \log_b(m+n) \). There is no simplification for the log of a sum.
- Mistake 2 — Ignoring the domain restriction: Applying \( \log \) to a negative number or zero is undefined. Always verify that the argument is strictly positive before proceeding.
- Mistake 3 — Misapplying the power rule: Writing \( \log_b(m^n) = (\log_b m)^n \) is a very common error. The correct form is \( \log_b(m^n) = n \cdot \log_b m \). The exponent becomes a coefficient, not another exponent.
- Mistake 4 — Confusing \( \log_b b \) and \( \log_b 1 \): Students sometimes mix these up. Remember: \( \log_b b = 1 \) always, and \( \log_b 1 = 0 \) always, regardless of the base.
- Mistake 5 — Wrong change-of-base direction: The change-of-base formula is \( \log_b m = \frac{\log_c m}{\log_c b} \). Students frequently invert the fraction, writing \( \frac{\log_c b}{\log_c m} \). The argument goes in the numerator and the base goes in the denominator.
JEE/NEET Application of Logarithm Formula
In our experience, JEE aspirants encounter logarithm formulas in at least 2-3 questions per paper, spanning algebra, coordinate geometry, and calculus. NEET aspirants meet logarithms in Chemistry (pH calculations, rate laws) and occasionally in Physics (decibel scale, radioactive decay). Below are the most important application patterns.
Pattern 1 — Solving Exponential Equations (JEE Main)
JEE Main frequently asks students to solve equations like \( 2^x = 7 \). Apply \( \log_{10} \) to both sides:
\[ x = \frac{\log_{10} 7}{\log_{10} 2} = \frac{0.8451}{0.3010} \approx 2.807 \]
This is a direct application of the change-of-base rule. Mastering this pattern alone can secure 4 marks in JEE Main.
Pattern 2 — Logarithmic Inequalities (JEE Advanced)
JEE Advanced tests the monotonicity of logarithm functions. If \( b > 1 \), then \( \log_b x \) is increasing. If \( 0 < b < 1 \), it is decreasing. This flips the inequality sign during solving. For example, solving \( \log_{0.5} x > 2 \) gives \( x < (0.5)^2 = 0.25 \) because the base is less than 1.
Pattern 3 — Chemistry pH and Rate Laws (NEET)
In NEET Chemistry, the pH formula is:
\[ \text{pH} = -\log_{10}[\text{H}^+] \]
The Arrhenius equation and integrated rate laws use \( \ln \). Students must convert using \( \ln x = 2.303 \log_{10} x \). Our experts suggest practising at least 10 pH and rate-law problems using logarithm formulas to build exam-day speed.
FAQs on Logarithm Formula
For more related formulas, explore our Complete Math Formulas hub. You may also find our articles on the De Moivre's Formula and the Asymptote Formula helpful for building a strong foundation in advanced algebra and complex numbers. For geometry topics, visit our guide on the Triangular Pyramid Formula. For the official NCERT syllabus, refer to ncert.nic.in.