🔢 Quant

Exponents, Surds & Logarithms

Laws of indices, surds, and logarithms.

of Quant

Why This Topic Matters

Total PYQs📊

of 1002 · 2021–2025

Years featured📅

4/5

of recent CAT years

% of Quant📈

~5%

of section questions

Est. hours⏱️

~8h

to master

~2/22

2021

2022

~2/22

2023

~2/22

2024

1/22

2025

🎯PYQ Evidence

CAT 2021–2025: ~1.1 per slot (2021: 1.3 · 2023: 1.7 · 2024: 1.7 · 2025: 1.0). Logarithms appear in four of five years (skipping only 2022); surds & indices clustered in 2023–24.

Indices, Surds & Logarithms

Three faces of the same idea — repeated multiplication. Master the laws and most questions collapse to one or two steps.

Laws of exponents

$a^m\cdot a^n=a^{m+n},\quad \frac{a^m}{a^n}=a^{m-n},\quad (a^m)^n=a^{mn},\quad a^0=1,\quad a^{-n}=\frac1{a^n}.$

Fractional powers are roots: $a^{1/n}=\sqrt[n]{a}$ , so $a^{m/n}=\sqrt[n]{a^m}$ .

Logarithms — the inverse

$\log_a N=x$ means $a^x=N$ . The three working laws:

$\log(xy)=\log x+\log y,\quad \log\frac xy=\log x-\log y,\quad \log(x^n)=n\log x,$

plus change of base $\log_a N=\dfrac{\log N}{\log a}$ and the identity $a^{\log_a N}=N$ .

A worked example

Solve $2^{\,x+2}=32$ .

Write both sides as powers of the same base:

$2^{\,x+2}=2^5\ \Rightarrow\ x+2=5\ \Rightarrow\ x=\mathbf{3}.$

Equating exponents works whenever you can match the base — the first move in almost every indices question.

Surd handling

Rationalise by multiplying by the conjugate:

$\frac{1}{\sqrt3-1}=\frac{\sqrt3+1}{(\sqrt3-1)(\sqrt3+1)}=\frac{\sqrt3+1}{2}.$

🎯PYQ Evidence

Force everything onto one base — then it's just algebra. : change-of-base rewrites log_100 x and log_1000 x as 1/2 and 1/3 of log_10 x, so with t = log_10 x the whole thing collapses to one linear equation; solving and taking the floor gives 31. : the bases 64, 8, 512 are all powers of 2, so converting to base 2 combines the logs into a single condition on log_2(xyz), fixing xyz — after which AM-GM (arithmetic mean ≥ geometric mean) hands you the smallest x+y+z = 48. : with A = 5x+9 and B = 5x−9, squaring √A + √B uses (√A+√B)² = A + B + 2√(AB); since A+B = 10x is rational, matching the plain and irrational parts pins x, giving √(10x+9) = 3√7. Pick the common base or square the surd, and the unknowns drop out.

Common traps

Adding exponents across a sum. $a^m+a^n$ does not simplify — only products do.
$\log$ of a sum. $\log(x+y)\ne\log x+\log y$ .
Negative/zero domain. $\log$ is defined only for positive arguments; check before applying laws.