Quant
Time, Speed & Distance
Every CAT Time, Speed & Distance formula on one page — relative speed, trains crossing poles and platforms, boats and streams, and delay problems — each with a worked example.
View
Logarithm
CAT'26 QUANT CHEATSHEET
Every logarithm formula and CAT shortcut you need for CAT 2026 — on one page.
Logarithms on CAT are rarely tested in isolation — they show up buried inside digit-counting questions, equations that need a domain check, and change-of-base chains that look intimidating until you spot the telescoping pattern. Treat every log as a hidden exponent and most of the topic reduces to matching the right law to the right shape. This sheet lays out every formula you need for the CAT quant section: the basic definition, the three log laws, special values, change of base, log equations and inequalities, digit counting via characteristic and mantissa, and the log-graph and inverse-function properties, each with a worked example in real numbers. Keep it open while you practise, and after a mock check where you stand on the CAT score predictor to see which idea is costing you marks.
Dev note: the canonical route
/cheatsheets/optima-learn-logarithm-cheatsheet is not live yet
(no /cheatsheets hub in the current sitemap). Built production-ready as a drop-in once it ships.Logarithm: every formula you need
1Logarithm Basics
The exponent a base must be raised to, to get a number.
logb x = y ⇔ by = x (b>0, b≠1, x>0)
Example: log2 32 = 5, since 25 = 32.
CAT Insight: Think of logarithms as hidden exponents — the log is just asking 'what power?'.
2Log Laws
Product, quotient and power rules simplify combined logs.
logb(MN) = logbM + logbN | logb(Mk) = k·logbM
Example: log28 + log24 = log232 = 5.
CAT Hack: Product → add; quotient → subtract; power → multiply the exponent out front.
3Special Values
A handful of log identities that recur constantly.
logb1 = 0 | logbb = 1 | logb(bk) = k
Example: log3(1/81) = log3(3−4) = −4.
CAT Insight: If the argument is already a power of the base, the answer is simply that exponent.
4Change of Base
Convert between any two bases using a third.
logab = logcb / logca = 1 / logba
Example: log23 · log35 · log58 = log28 = 3.
CAT Hack: Remember logab · logba = 1 — the chain always telescopes back to one log.
5Log Equations
Combine the logs first, then convert to exponential form.
logbf(x) = k ⇔ f(x) = bk — always check the domain
Example: log2(x−1) + log2(x+1) = 3 → x2=9 → x = 3 (x=−3 rejected).
Common Mistake: Always reject values that make any log argument ≤ 0, even if they solve the equation algebraically.
6Log Inequalities
The direction of the inequality depends on the base.
b>1: inequality unchanged | 0<b<1: inequality reverses
Example: log0.5x > 2, base < 1 so reverse → 0 < x < 0.25.
CAT Hack: Base greater than 1 preserves direction; base between 0 and 1 flips it.
7Digit Counting
The number of digits in a positive integer, from its log.
digits in N = ⌊log10N⌋ + 1
Example: digits in 2100, log 2 = 0.301 → 100·0.301 = 30.1 → 31 digits.
CAT Favourite: One formula solves almost every digit-count problem on CAT — just floor the log and add 1.
8Characteristic & Mantissa
Every common logarithm splits into an integer and decimal part.
log10N = characteristic + mantissa, 0 ≤ mantissa < 1
Example: log(250) = 2.398 → characteristic = 2, mantissa = 0.398.
CAT Insight: Only the characteristic decides the digit count; the mantissa never affects it.
9Change of Base Applications
Chain-rule identities that collapse combined logs fast.
1/logab = logba | logab·logbc = logac
Example: 1/log236 + 1/log336 = log362+log363 = log366 = ½.
CAT Hack: Convert awkward logs to their reciprocal form, or chain them, before trying to simplify directly.
10Log Graph & Properties
The shape and asymptote of y = log base b of x.
domain x>0 | passes through (1,0) and (b,1) | vertical asymptote at x=0
Example: b > 1 gives an increasing curve; 0 < b < 1 gives a decreasing curve, both through (1, 0).
CAT Hack: A smaller base above 1 gives a larger log value for the same x > 1.
11Log & Exponential Relation
Logarithm and exponential are inverse functions of each other.
y = logbx is the inverse of y = bx (graphs reflect about y = x)
Example: blogb7 = 7 and logb(bx) = x.
CAT Hack: Use the inverse property directly to cancel a log and an exponential in the same base.
CAT exam shortcuts, traps & revision
12
CAT Exam Shortcuts
- Argument of a log must be > 0; base must be > 0 and ≠ 1
- logb(MN) = add; logb(M/N) = subtract; logb(Mk) = multiply by k
- Change of base: logab = logcb/logca; logab·logba = 1
- Always check the domain in log equations and inequalities
- Inequality reverses only when the base is between 0 and 1
- Digits in N = ⌊log10N⌋ + 1; only the characteristic decides digit count
13
Most Common CAT Traps
- Forgetting to reject solutions that make a log argument zero or negative.
- Applying log laws without checking the domain restrictions first.
- Not reversing a log inequality when the base is between 0 and 1.
- Mixing up characteristic and mantissa when counting digits.
- Forgetting that a smaller base above 1 gives a larger log value.
14
30-Second Revision Box
- logbx = y ⇔ by = x; b>0, b≠1, x>0
- logb1=0; logbb=1; logb(bk)=k
- Change of base: logab = logcb/logca
- Base > 1 preserves inequality direction; 0<base<1 reverses it
- Digits in N = ⌊log10N⌋ + 1
- Log and exponential are inverses; graphs reflect about y = x
Logarithms reward recognising the shape of an expression over grinding through algebra — once you see a change-of-base chain or a hidden power of the base, the answer is usually one line away. Drill this sheet until the log laws and the digit-counting formula are reflex, then test them on full sets and track progress with the CAT score predictor. For more topic guides, browse the Optima Learn blog or explore every study guide, and work through the full CAT exam hub for section-wise strategy. When you want structured, mentor-led prep, the team at Optima Learn can map out your plan — book a free CAT 2026 call and line up your next eight weeks.
