CAT Logarithms, Surds and Indices: The 6 Rules to Know

Quant

Quick question: which is larger, the square root of 2 or the cube root of 3? Most aspirants squint at the calculator in their head and guess. A single rule settles it in ten seconds, because raising both numbers to the sixth power turns the question into 8 versus 9.

That small trick is the whole spirit of this chapter. Indices, surds and logarithms look like three separate topics with a pile of formulas. They are really one toolkit for rewriting numbers into a form you can compare or simplify. Six rules do most of the work, and this guide walks through each with a worked case.

Why Indices, Surds and Logarithms Belong Together

A logarithm is an index in disguise. To say that log of 8 to the base 2 is 3 is to say that 2 raised to 3 is 8. A surd is also an index: the square root of 5 is 5 raised to one half. Once you see all three as exponent questions, the chapter stops feeling like three.

That matters for how you prepare. Aspirants who treat each topic as its own island tend to memorise twenty formulas and use four. Aspirants who see the single idea underneath tend to rebuild the rest on demand. CAT rewards the second group, because the questions rarely match a template exactly. Quant overall has historically leaned most on arithmetic and algebra, and this chapter is a compact, high return part of algebra.

These questions also tend to be short. The setup is usually a line or two, and the difficulty is choosing the right rewrite. That makes them valuable for anyone managing time in the Quant section, since a question you recognise costs a minute while one you do not can eat four.

The six rules below are ordered from the easiest to the one people skip. Learn them as a ladder: each step builds on the one before, and a question usually needs only one or two rungs.

The Six Rule Ladder: What to Know Cold

Keep these six lines in your head. The sections that follow explain when each fires and how to apply it without slipping.

Notice how the rungs connect. Rule one feeds rule two, because you cannot equate exponents until the laws have helped you reach a single base. Rule five feeds rule six, because a combined log is what you convert into an index form. Rules three and four sit between them and handle the roots. Each rung is short, and the ladder is the sequence you walk through when a question will not simplify.

Think of the ladder as a checklist rather than a syllabus. On a fresh question, ask three things in order. Are the bases the same, or can they be made the same? Is there a root that needs clearing or denesting? Is there a log whose arguments I can combine? One of the three nearly always says yes.

Indices: Reduce to One Base Before Anything Else

Rules one and two cover indices. When bases are the same, exponents add under multiplication and subtract under division. Zero as an exponent gives 1, and a negative exponent flips the base into a reciprocal. These facts are old friends, and the aim is to use them without slowing down.

The real leverage comes from rule two. Suppose 4 to the power (x plus 1) equals 8 to the power (x minus 1). Rewrite 4 as 2 squared and 8 as 2 cubed. The equation becomes 2 to the power (2x plus 2) equals 2 to the power (3x minus 3), so 2x plus 2 equals 3x minus 3, and x is 5. A single base turned a messy equation into a line of algebra.

The same rule settles comparisons. To compare the square root of 2 with the cube root of 3, raise both to the sixth power. You get 2 cubed, which is 8, against 3 squared, which is 9. The cube root of 3 is larger. Whenever roots of different orders appear, use the least common multiple of the orders.

Watch for the edge cases. If a to the power x equals b to the power x, the bases are equal only when x is non zero, and sign matters when the exponent is even. Questions that test this are rare but they exist, so keep it in mind when a cleaner looking option seems too easy.

Two more patterns are worth a line each. Powers of a number with fractional exponents, such as 8 to the power two thirds, split into a root and a power: take the cube root first to get 2, then square to get 4. Doing the root first keeps the numbers small and cuts the chance of an arithmetic slip, which is the usual cost in these questions.

Finally, remember that a negative exponent only moves the base across the fraction bar. It never makes the value negative. Many options in this chapter include the negated version of the right answer as bait, so a quick sign check against the original expression is worth the three seconds it takes.

Surds: Rationalise and Denest

Surds questions usually ask you to simplify, compare or evaluate an expression with square roots. Rule three is rationalising. To clear a root from a denominator, multiply top and bottom by the conjugate. For 1 over (3 plus the root of 5), multiply by (3 minus the root of 5). The denominator becomes 9 minus 5, which is 4, and the answer is (3 minus the root of 5) over 4.

Rule four is denesting. An expression like the root of (7 plus 4 root 3) looks stuck. Ask whether it equals a plus root b, where squaring gives a squared plus b plus 2a root b. Matching terms with 7 plus 4 root 3 suggests 2 plus root 3, because 4 plus 3 is 7 and twice 2 times root 3 is 4 root 3. The check is immediate.

A good habit is to write the expression as the root of (p plus 2 root q) and look for two numbers that add to p and multiply to q. For 7 plus 4 root 3 that is 7 plus 2 root 12, and the numbers 4 and 3 add to 7 and multiply to 12. This turns denesting into a quick search over factor pairs.

Conjugates have a second use. If x is 2 plus the root of 3, then 1 over x is 2 minus the root of 3, because the product of the two is exactly 1. That makes x plus 1 over x equal to 4, and x squared plus 1 over x squared equal to 14. Questions that give a surd and ask for such a symmetric expression reward this rule and need almost no arithmetic.

Logarithms: The Laws, the Domain and Digit Counting

Rule five is the set of log laws. The log of a product is the sum of logs, the log of a quotient is the difference, and the log of a power is the power times the log. Add change of base, where log of b to the base a equals one over the log of a to the base b. For example, log of 8 to the base 4 is 3 over 2, because both numbers are powers of 2.

Rule six is the one aspirants forget. Solve log to the base 2 of x plus log to the base 2 of (x minus 2) equals 3. Combine to get x times (x minus 2) equals 8, so x squared minus 2x minus 8 is zero and x is 4 or minus 2. Only 4 survives, because the argument of the second log must be positive. Skipping the domain check hands the wrong option a free victory.

Logs also count digits. The number of digits in N is the integer part of log of N, plus one. Take 2 to the power 50. Its log to base 10 is 50 times 0.3010, which is 15.05, so the number has 16 digits. That works only if you remember log of 2 as 0.3010 and log of 3 as 0.4771, so commit both to memory.

Rule	Cue in the question	First move
Same base laws	Powers multiplied or divided	Add or subtract exponents
Reduce to one base	Different bases, one equation	Rewrite as powers of a common base
Rationalise	Root in a denominator	Multiply by the conjugate
Denest	Root of a sum with a surd	Match to (a plus root b) squared
Log laws	Sum or difference of logs	Combine into one log
Domain check	A log equation with x inside	Reject any x making an argument non positive

A Practice Routine for the Six Rules

Rules stick through application, not rereading. For two weeks, solve four or five questions a day from this chapter and write the rule number beside each. The labelling forces you to name the move, and the move is what you will need to recall in the exam.

When you get a question wrong, ask whether the rule was unknown, known but not triggered, or misapplied. Each cause needs a different fix. An unknown rule needs revision, an untriggered rule needs more cue spotting, and a misapplied rule needs slower, written steps. Most errors fall in the second and third groups.

Calculation fluency helps here too. Our piece on memorising squares and cubes beyond 30 makes denesting and base reduction faster, since you spot perfect squares and cubes on sight. For choosing between a clean algebraic route and plugging in options, see algebraic versus numerical methods in CAT Quant. Then test the surd side with surds and indices practice questions for CAT.

Options in these questions can be deliberately tempting. Read how CAT Quant options include obvious trap values so that the sign mistake or the skipped domain check does not cost you a clean question. For the time side of the story, see time consuming versus difficult CAT Quant questions.