Logarithms for CAT: Concepts & PYQs

Logs, exponents, surds and indices recur in recent CAT papers and are often mixed with equations/inequalities. For CAT preparation, the useful question is not “Have I finished the chapter?” but can I recognise the version of the concept CAT hides inside a word problem and choose an efficient method under time pressure?

What to know

The core checklist for this topic is: Definition, laws, change of base, exponential-log conversion, domains, common bases; and log inequalities, indices/surds. Learn these as connected ideas rather than isolated formulas. CAT often combines two of them in one question.

Recurring CAT-style forms

Expect questions built around Solve log equation, compare logarithms, parameter/domain, nested logs, exponential growth, and log with polynomial argument. The wording changes, but the mathematical structure repeats. Your first 20–30 seconds should be spent identifying that structure before writing equations.

Decision rules worth remembering

• log(ab)=log a+log b
• log(a/b)=log a−log b
• log(a^k)=k log a
• base>0, base≠1, argument>0
• change-of-base

One original example

log₂(x−1)+log₂(x−3)=3 becomes (x−1)(x−3)=8, but the domain first requires x>3. Solve only after protecting the domain.

The point of a worked example is not the answer; it is the choice of representation. After solving, ask whether a ratio table, base-100 assumption, sign chart, factorisation or diagram made the work shorter.

Common traps

• log(a+b)≠log a+log b
• ignoring positive argument
• base between 0 and 1 reverses inequality
• canceling logs with different bases

These traps are valuable because they explain why a student can “know the formula” and still lose marks. Add the exact mistake to your error log when it occurs; do not simply write “silly mistake.”

How to practise

Start with 10–15 untimed questions until the first step becomes automatic. Then use mixed timed sets where this topic is not announced in advance. Finish with sectionals and mock review. Mastery checkpoint: Checks domain first automatically and can switch between exponential/log form fluently.

Once that checkpoint is stable, move on rather than over-polishing one chapter. CAT QA is a portfolio of scoring areas; the objective is enough reliable topics to make question selection easy on exam day.