Logarithm Formulas for CAT 2026: 18-Formula Cheatsheet
Logarithm formulas for CAT 2026 occupy a tiny corner of the syllabus but carry an outsized weight on the actual paper. The whole topic runs to about 18 formulas, yet it has appeared in roughly seven out of every ten CAT papers in the past decade. This cheatsheet pins every formula to a CAT use case and ends with 15 must-solve questions that mirror the patterns CAT setters actually use, so the topic moves from "memorised" to "recognised in 60 seconds" by the end of one focused weekend.
The reason most aspirants lose marks on log questions is not formula gaps. The cheatsheet on the internet usually lists the right formulas. The gap is recognition speed: knowing which formula collapses a question in 60 seconds instead of three minutes. This blog fixes that by linking each formula to the question type it solves and removing the cheatsheet-to-paper translation problem entirely.
Why Logarithm Formulas Matter More Than the Syllabus Suggests
Logarithms occupy a tiny corner of the CAT Quantitative Aptitude syllabus, getting just one sentence in most syllabus PDFs. Yet on the actual paper, log questions appear with surprising regularity, usually as a clean two-mark question or as a hidden component inside an algebra or inequality problem, which makes the topic a much larger contributor to scores than its syllabus footprint suggests.
The return on effort here is unusually strong. A weekend mastering 18 formulas covers more scoring ground than a weekend on, say, Geometry. Logarithms also sit underneath several adjacent topics in disguise: compound interest with continuous compounding leans on the natural log, inequality questions involving exponential growth reduce to log comparisons, and certain Number System questions are logarithmic in form. Strong log fluency raises performance across multiple Quant areas at once.
The 18 Logarithm Formulas for CAT 2026 Cheatsheet
The cheatsheet groups formulas into four blocks. Each block has a recognition cue: the kind of question pattern that signals the block is in play. Working through them in this order, instead of as a flat list, helps the brain reach for the right formula faster on exam day.
Block 1 — Core Log Properties (4 formulas)
The four properties form the foundation of every log calculation. Almost every CAT log question uses at least one of these, often two stacked together. The recognition cue: any expression with logs being added, subtracted, or multiplied by an exponent, or a log of a product, quotient, or power.
| # | Formula | Read it as |
|---|---|---|
| 1 | logb(xy) = logbx + logby | Log of a product is sum of logs. |
| 2 | logb(x/y) = logbx − logby | Log of a quotient is difference of logs. |
| 3 | logb(xn) = n · logbx | The exponent steps out front. |
| 4 | logb1 = 0 | Log of one is always zero. |
Block 2 — Change of Base and Reciprocals (4 formulas)
If the four core properties are the foundation, change of base is the lever. CAT questions that look impossible in one base often collapse into trivial arithmetic when rewritten in another base. The recognition cue: a question with logs in two different bases, or a log expression that needs to be evaluated numerically.
| # | Formula | Use case |
|---|---|---|
| 5 | logbx = logkx / logkb | Rewrite any log in any base of your choice. |
| 6 | logbx = 1 / logxb | Swap base and argument by taking reciprocal. |
| 7 | logab · logba = 1 | Product of swapped-base logs is one. |
| 8 | logab · logbc · logca = 1 | Three-way log chains telescope to one. |
Block 3 — Special Values and Cancellation (5 formulas)
These five identities are the fastest answer keys in the topic. When a question matches one of these patterns, the answer is often visible in one line. The recognition cue: a log whose base and argument share a structural relationship.
| # | Formula | Pattern |
|---|---|---|
| 9 | logbb = 1 | Log of base itself. |
| 10 | logb(bn) = n | Log of base raised to n. |
| 11 | blogbx = x | Base raised to log of x cancels. |
| 12 | log(bn)(xn) = logbx | Same power in base and argument cancels. |
| 13 | log(bn)x = (1/n) · logbx | Power in base steps out as reciprocal. |
Block 4 — Advanced and Mixed Identities (5 formulas)
The final five formulas are the ones CAT setters reach for when designing TITA questions, where there are no answer options to reverse-engineer. The recognition cue: an expression that combines exponentials with logs, or a log inside a log.
| # | Formula | Where CAT uses it |
|---|---|---|
| 14 | alogbc = clogba | Swap base and argument of the exponent. |
| 15 | logb(bx) = x and blogbx = x | The two-way inverse of base b and log b. |
| 16 | If logbx = logby then x = y | Same-base equality drops the log. |
| 17 | logbx > 0 iff x > 1 (b > 1) | Sign of log tells you which side of 1. |
| 18 | log102 ≈ 0.301; log103 ≈ 0.4771 | The two values worth memorising. |
Domain Rules That Save You From Wrong Answers
Two domain rules apply to every log expression on a CAT paper, and skipping these is the single most common reason log questions are answered with extraneous solutions that look right but are not. Rule one: the argument must be strictly positive. The expression logbx is only defined when x is greater than zero, so if a log equation produces a solution that makes the argument zero or negative, that solution is invalid and must be dropped.
Rule two: the base must be positive and not equal to one. The expression logbx requires b greater than zero and b not equal to one. The base b equals one is undefined because every power of one is one. These two rules sound trivial yet are responsible for an outsized share of wrong answers on TITA log questions, where there is no answer option to sanity-check against. Building the habit of checking domain before submitting is worth a real marks-per-paper lift across mocks and the actual CAT paper.
15 Must-Solve CAT Logarithm Questions
These 15 questions cover the four blocks above, each tagged with the formula or trap it tests. The drill: see the question, identify the relevant block, solve in under 90 seconds.
If log102 = 0.301, find log1080.
log10(8 × 10) = 3(0.301) + 1 = 1.903. Answer: 1.903
Simplify log10(125/8) using log102 = 0.301.
3 log105 − 3 log102 = 3(0.699) − 3(0.301) = 1.194. Answer: 1.194
Find log2(645).
log2(645) = 5 log264 = 5 × 6 = 30. Answer: 30
Evaluate log432.
log432 = log232 / log24 = 5/2 = 2.5. Answer: 2.5
Find log23 · log34 · log45 · log56 · log67 · log78.
The chain telescopes by change of base: each ratio cancels the next, leaving log28 = 3. Answer: 3
If log2x = 5, what is logx2?
logx2 = 1 / log2x = 1/5 = 0.2. Answer: 0.2
Find the value of 5log513.
blogbx = x, so 5log513 = 13. Answer: 13
Find log981.
log981 = log9(92) = 2. Or, log(32)(34) = 4/2 = 2. Answer: 2
Evaluate log25625.
log(52)(54) = 4/2 = 2. Answer: 2
Show that 2log49 = 3.
2log49 = 2(log29)/2 = (2log29)1/2 = 91/2 = 3. Answer: 3
Solve log3(2x + 1) = log3(x + 7).
Same base, so 2x + 1 = x + 7, giving x = 6. Check domain: both arguments positive at x = 6. Answer: x = 6
If (log2x)2 − 3 log2x + 2 = 0, find x.
Let y = log2x. Then y2 − 3y + 2 = 0, so y = 1 or y = 2. So x = 2 or x = 4. Answer: x = 2 or 4
For which integer values of x in (1, 100) is log2x an integer?
log2x is an integer when x is a power of 2. In (1, 100), the powers of 2 are 2, 4, 8, 16, 32, 64. That is 6 values. Answer: 6 values
Solve log2(x − 3) + log2(x − 1) = 3.
Combine: log2((x − 3)(x − 1)) = 3, so (x − 3)(x − 1) = 8. Expand: x2 − 4x − 5 = 0, so x = 5 or x = −1. Reject x = −1 (argument negative). Answer: x = 5
If log1227 = a, express log616 in terms of a.
Set a = 3 log123, so log123 = a/3. By change of base in base 2, log23 = 2a/(3 − a). Then log616 = 4/(1 + log23) = 4(3 − a)/(3 + a). Answer: 4(3 − a) / (3 + a)
How CAT 2026 Logarithm Questions Will Likely Differ From 2025
Three patterns from recent CAT papers are worth flagging for CAT 2026 aspirants. First, log questions are appearing more often as a component inside an algebra or quadratic problem, not as standalone log problems. A question is set up as a quadratic, and the unknown turns out to be a log expression, which rewards students who can spot a log substitution. Second, TITA log questions are using domain traps more aggressively. The setter writes an equation that produces two algebraic solutions, expecting candidates to forget that one solution makes a log argument negative.
Third, the change of base formula is appearing in disguise. A question may not look like a base change question, but solving it cleanly requires rewriting one log in another base. Recognising this pattern is the single highest-value drill in the topic. A focused log preparation plan should spend more time on these three patterns than on standalone formula drills, and the CAT practice questions library on Optima Learn surfaces log problems tagged by exactly these patterns.
Get a Personalised CAT 2026 Quants Roadmap
Logarithms is one of 22 topics that need to slot into your CAT 2026 Quants plan. A diagnostic-driven roadmap maps your current standing to which topics deserve the next two weekends of your time.
Map My CAT 2026 Quant PlanWhere Logarithm Formulas for CAT Fit in the 9-Month Plan
A common error in CAT 2026 preparation is studying topics by alphabetical or syllabus order rather than by return on effort. Logarithms should sit early in any plan because the effort-to-yield ratio is unusually good and the topic can be mastered in a focused weekend. Inside a 9-month plan, a sensible placement is in month one alongside other small, high-ROI Quants topics like AP, GP, surds, and indices. By month two, time freed up can be redirected to higher-volume topics like Number System and Arithmetic, and mocks from month two onwards already test something the candidate can answer, which lifts mock score morale early.
For a working professional or a college student doing CAT alongside other commitments, logarithms suits short study blocks of forty minutes each. For a fuller view of how to sequence Quants topics across a 9-month plan, our CAT exam guide walks through the trade-offs by aspirant type, and the CAT 2026 waitlist details page explains how the personalised planner builds your topic sequence.
Three Habits That Make Log Questions Feel Easy
Once the 18 formulas are in memory, three habits separate aspirants who solve log questions in under 60 seconds from those who take three minutes. Habit one: write the recognition cue first. Before reaching for any formula, write a single word on the rough sheet that names the block in play: properties, base-change, special, or advanced. This forces a category match before formula choice and cuts wrong-formula attempts.
Habit two: rewrite to base 2 by default. When a question has logs in multiple bases or an awkward base like 4 or 8, rewriting everything to base 2 usually simplifies the arithmetic. Habit three: domain check before submitting. For any log equation that produced two algebraic roots, plug both back into every original log argument and reject any root that makes any argument zero or negative. This habit alone is worth one to two marks per paper on average. These habits install through timed practice, ideally as part of a chapter-by-chapter learning sequence. The CAT blogs library on Optima Learn has companion cheatsheets for AP GP, Quadratic Equations, and Number System that follow the same recognition-first structure used here.
Common Doubts About CAT Logarithm Preparation
Do I need to know natural logarithms for CAT?
CAT does not test natural logarithms directly. Almost every CAT log question uses base 10 or base 2, sometimes base 3 or base 5. The natural log (base e) shows up only in adjacent topics like continuous compound interest, and even there the question can usually be solved without invoking e. A working knowledge of common logarithms is sufficient for CAT.
How many log questions should I expect in CAT 2026?
One to two direct log questions in a typical CAT QA section, plus one to two adjacent questions where a log identity is the cleanest path. That is three to four marks of expected log content per paper. Given that 18 formulas cover the topic, that is roughly one mark per four formulas memorised, which is a strong return on the weekend of preparation.
How do I revise logarithms one week before CAT 2026?
A one-week revision plan: day one, re-read the 18 formulas and solve five recognition drills (one per block). Day two to four, attempt one full set of 10 log questions under timed conditions and review errors. Day five to six, attempt CAT-level log questions from previous year papers. Day seven, scan only the cheatsheet for a 15-minute final review before the exam.
What if I keep getting log inequality questions wrong?
Log inequality questions are the hardest sub-pattern of the topic. The fix is the sign rule (formula 17 above) plus careful handling of when to flip the inequality, which happens when the base is less than 1. Drill five inequality questions a day for a week with a strict habit of writing the base case explicitly before solving. That clears most of the recurring errors.
Final note. Logarithms is the rare CAT topic where a complete weekend of focused work produces a permanent skill. Eighteen formulas, four blocks, three habits, and fifteen practice questions are the whole topic. After that weekend, the only remaining task is to keep the recognition reflex sharp through occasional drill questions, which the CAT score predictor and full-length mocks already provide.
