Logarithms
A logarithm answers one question: to what power must we raise the base to get a given number? Written log_b N = x, it simply means b^x = N — so a log is just an exponent wearing a different hat. That single idea makes logarithms a quiet workhorse in CAT Quant: they turn multiplication into addition, division into subtraction, and powers into multiplication, which is exactly why they crack problems that look impossible by direct computation. CAT does not test heavy log tables; it tests whether you can manipulate the three core laws fluently, switch bases when a question mixes log_2 and log_3, solve clean logarithmic equations, and use the characteristic of log_10 to count the digits of a giant number like 5^100. This chapter builds that fluency in order — the laws first, then change of base, then equations, then the characteristic-and-mantissa digit trick — each with worked CAT-style examples, the fastest method, and the traps (domain errors, dropped bases) that quietly cost marks.
Topics
⚡ CAT shortcuts & speed methods
The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.
- Read log_b N as "the power on b that gives N". log_2 8 = 3 because 2^3 = 8 — no formula needed.
- A coefficient in front of a log is a hidden power: 3·log 2 = log 8, ½·log 36 = log 6. Use it to merge terms.
- Change of base to flip: log_b a = 1/log_a b. A chain log_2 3·log_3 4·…·log_(n−1) n telescopes to log_2 n.
- Base-power rule: log_(b^m)(a^n) = (n/m)·log_b a — slide powers between base and argument (log_8 32 = 5/3).
- For log equations, combine to one log, convert log_b X = k to X = b^k, then ALWAYS check the argument is positive.
- Digit count (base 10): digits of N = ⌊log_10 N⌋ + 1. Keep log 2 ≈ 0.30103, log 3 ≈ 0.47712, log 5 = 1 − log 2.
⚠️ Common mistakes & traps
CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Treating log(M + N) as log M + log N — the laws only apply to products, quotients and powers, never to a sum.
- Combining logs with different bases without first applying change of base.
- Accepting an algebraic root that makes a log argument (or base) zero or negative — extraneous roots must be rejected.
- Confusing the digit formula: it is ⌊log N⌋ + 1, not ⌈log N⌉ — taking the ceiling overcounts on exact powers of 10.
- Mishandling negative characteristics: for N < 1, log N = −2.3 means characteristic −3 (not −2), since the mantissa stays non-negative.
📈 CAT exam insight & PYQ analysis
🎴 Flashcards — instant recall
Tap a card to reveal the answer. Drill these until they are automatic.
📌 Quick revision
Chapter test
🏆 Vidaara CAT success checklist
You have truly mastered Logarithms when you can tick every box below.
- Recall every formula in this chapter without looking them up
- Solve each topic’s practice set with at least 80% accuracy
- Use the chapter shortcuts to cut your solving time in half
- Spot and avoid every common trap listed above
- Score 80%+ on the timed chapter test
📋 Chapter mastery scorecard
Track where you stand. Aim for the target before moving to the next chapter.
| Skill checkpoint | Target |
|---|---|
| Concept theory & formulas understood | 100% |
| Topic practice sets attempted (4 topics) | 4/4 |
| Best topic-test score | — → 80%+ |
| Chapter test score | — → 80%+ |
| Flashcards drilled to instant recall | 12 cards |
Formula Reference Sheet
The three laws
| Definition | log_b N = x ⇔ b^x = N (b > 0, b ≠ 1, N > 0) |
|---|---|
| Product law | log_b(MN) = log_b M + log_b N |
| Quotient law | log_b(M/N) = log_b M − log_b N |
| Power law | log_b(M^n) = n · log_b M |
| Log of base / of 1 | log_b b = 1, log_b 1 = 0 |
CAT power-tools
| Change of base | log_b a = (log a)/(log b) = (log_c a)/(log_c b) |
|---|---|
| Reciprocal rule | log_b a · log_a b = 1, i.e. log_b a = 1/log_a b |
| Base-power rule | log_(b^m)(a^n) = (n/m) · log_b a |
| Identity | b^(log_b N) = N and a^(log_b c) = c^(log_b a) |
| Digit count of N (base 10) | digits = ⌊log_10 N⌋ + 1 |