Indices, Surds and Logarithms Notes for Competitive Exams

Indices, surds and logarithms form a basic part of the quantitative aptitude section in SSC, banking, railways, defence and state-level examinations. The questions are short and based on a few rules, so they are among the quickest to solve if you know the formulas. A candidate who learns the laws of indices, the methods to simplify surds and the main rules of logarithms can answer such questions in less than a minute each. This note explains all these topics in simple language with formulas, solved examples and practice questions, and also gives the value tables that save time in the exam.

Part 1: Indices (Exponents)

When a number is multiplied by itself several times, we write it as a power: a x a x a = a³. Here a is the base and 3 is the index or exponent. Indices make the calculation of large numbers easy.

Laws of Indices

If the bases are equal in an equation, the indices must be equal. If a^x = a^y, then x = y. For example, 2^x+1 = 32 means 2^x+1 = 2⁵, so x + 1 = 5, and x = 4. Similarly, 3^2x = 81 = 3⁴ gives 2x = 4, so x = 2.

Part 2: Surds

A surd is an irrational root of a rational number, such as the square root of 2 or the cube root of 5. The square root of 4 is not a surd, because it is equal to 2, a rational number. Surds cannot be written as exact decimals, so we keep them in the root form while simplifying.

Laws of Surds

• The square root of a x the square root of b = the square root of (a x b).
• The square root of a / the square root of b = the square root of (a / b).
• (The square root of a) x (the square root of a) = a.
• (a + the square root of b)(a - the square root of b) = a² - b. The two brackets are called conjugates.

Rationalisation

To remove a surd from the denominator, multiply the numerator and the denominator by the conjugate of the denominator. Example: 1 / (the square root of 3 - the square root of 2). Multiply by (the square root of 3 + the square root of 2). The denominator becomes 3 - 2 = 1, so the answer is the square root of 3 + the square root of 2. Another example: (the square root of 5 + the square root of 3) / (the square root of 5 - the square root of 3) = (the square root of 5 + the square root of 3)² / (5 - 3) = (5 + 3 + 2 times the square root of 15) / 2 = 4 + the square root of 15.

Square Root of a Quantity of the Form a + 2 root b

To find the square root of a surd expression, write it as (x + y) + 2 times the square root of (xy), where x + y = a and xy = b. For instance, to find the square root of (7 + 4 times the square root of 3): write it as 7 + 2 times the square root of 12. We need two numbers whose sum is 7 and product is 12, which are 4 and 3. So the square root is (the square root of 4 + the square root of 3) = 2 + the square root of 3. You can check: (2 + the square root of 3)² = 4 + 3 + 4 times the square root of 3 = 7 + 4 times the square root of 3.

Comparing Surds

To compare two surds with different orders, make the orders equal by taking the LCM of the orders. For example, compare the square root of 3 and the cube root of 4. The LCM of 2 and 3 is 6. The square root of 3 = 3^3/6 = 27^1/6, and the cube root of 4 = 4^2/6 = 16^1/6. Since 27 is greater than 16, the square root of 3 is greater.

Part 3: Logarithms

A logarithm is the inverse of an exponent. If a^x = N, then log base a of N = x. For example, since 2⁵ = 32, log base 2 of 32 = 5. The base a must be positive and not equal to 1, and N must be positive. Logarithms with base 10 are called common logarithms, and logarithms with base e (about 2.718) are natural logarithms, written as ln.

Laws of Logarithms

Useful values for the exam: log 2 = 0.3010, log 3 = 0.4771, log 5 = 0.6990 and log 7 = 0.8451 (to base 10). Example: log 16 = 4 x log 2 = 4 x 0.3010 = 1.2040. Another: log 5 = log (10/2) = 1 - 0.3010 = 0.6990. And log base 10 of 1000 = 3, because 10³ = 1000.

Squares, Cubes and Roots to Remember

Learn the squares up to 30 and the cubes up to 15. Some decimal roots can be written from them: the square root of 0.0016 = 0.04, and the cube root of 0.027 = 0.3.

Practice Questions With Answers

• Q1. Simplify (2³ x 3²) / (2² x 3). Answer: 6.
• Q2. Find the value of 8^2/3. Answer: 4.
• Q3. If log x = 2 (base 10), find x. Answer: 100.
• Q4. Find log base 3 of 81. Answer: 4.
• Q5. Rationalise 6 / (the square root of 3). Answer: 2 times the square root of 3.
• Q6. Find the value of (the square root of 2 + 1)(the square root of 2 - 1). Answer: 1.

Try to solve each question within 30 seconds. If you need more time, go back to the law that was used and write it in your notebook. After a week of daily practice, you will see that these questions become automatic, and you can then use the saved time on harder questions like data interpretation. In the SSC and banking examinations, one or two direct questions from indices and surds are common, and the topic also helps in solving equations and simplification questions in other chapters.

Tips and Common Errors

• Bring all terms to the same base before applying the laws of indices.
• Do not write (a + b)² as a² + b²; the correct form is a² + 2ab + b².
• In logarithms, log (m + n) is not log m + log n.
• Always rationalise the denominator before simplifying a fraction with a surd.

Revise related topics with our notes on number system and algebra.

Quick Revision

• a^m x aⁿ = a^m+n; a⁰ = 1; a^-n = 1/aⁿ.
• Conjugate pair: (a + root b)(a - root b) = a² - b.
• log(mn) = log m + log n; log(m/n) = log m - log n; log mⁿ = n log m.
• log 2 = 0.3010; log 3 = 0.4771.

Frequently Asked Questions (FAQ)

What is an index or exponent?

An index tells how many times a base is multiplied by itself, such as 2 to the power 3 equals 8.

What is a surd?

A surd is an irrational root of a rational number, like the square root of 2, which cannot be written as an exact decimal.

How do I rationalise the denominator?

Multiply the numerator and denominator by the conjugate of the denominator.

What are conjugate surds?

Two expressions like (a + root b) and (a - root b) are conjugates, and their product is a squared minus b.

What is a logarithm?

A logarithm is the power to which a base must be raised to get a number, so if a to the power x is N, then log N to base a is x.

What is log 1 equal to?

The log of 1 is 0 for any base.

What is the value of log 2 to base 10?

The value of log 2 to base 10 is about 0.3010.

Is log (m + n) equal to log m + log n?

No. The product rule is log (mn) = log m + log n, not for addition.

What is the value of any number to the power zero?

Any non-zero number raised to the power zero is equal to 1.

How do I compare two surds of different orders?

Take the LCM of their orders, convert both to the same order and compare the numbers under the root.

What are common logarithms?

Logarithms with base 10 are called common logarithms.

How can I improve speed in indices and surds?

Learn the laws, memorise squares and cubes, and practise the standard types of simplification daily.