All Logarithm Rules for NEET — Complete Reference

📏 1 · The Basics — what a logarithm is

A logarithm is just a tidy way to write the question "what power do I raise this number to?". It is the inverse of an exponential.

If a^x = N ⇔ log_a N = x

🎈 Plain English

"What power of a gives N?" — that answer is the logarithm of N to base a. So log₁₀ 1000 = 3 means "10 multiplied by itself 3 times gives 1000".

1.1 Three parts of a log expression

Base (a) — the number being multiplied. Must be positive and ≠ 1.
Argument (N) — the result you reach. Must be positive.
Log value (x) — how many times the base was multiplied. Can be any real number (positive, negative, zero, fractional).

1.2 The two bases NEET cares about

Name	Symbol	Base	Where it appears
Common log	`log N` or `log₁₀ N`	10	pH, pOH, pK_a, Nernst, decibels
Natural log	`ln N` or `log_e N`	e ≈ 2.718	Arrhenius, radioactive decay, exponential growth

📏 2 · Basic Identities

Rule	Formula	Plain meaning
Log of 1	`log_a 1 = 0`	Any base to the power 0 is 1.
Log of the base	`log_a a = 1`	Base raised to power 1 is itself.
Log of a base-power	`log_a(a^x) = x`	Read the exponent off directly — base "cancels".
Inverse identity	`a^{log_a N} = N`	Power and log are inverses — they undo each other.
One-to-one	If `log_a x = log_a y`, then x = y	Different inputs give different log values.
Equal bases & arguments	If `log_a x = log_b x` & x ≠ 1, then a = b	Equal logs at the same non-trivial argument force equal bases.

⚡ 10^{log x} = x and e^{ln x} = x. These two "cancellations" appear in 1 out of every 4 NEET kinetics or pH questions.

📏 3 · The Three Operation Rules (most useful!)

Rule	Formula	Plain meaning
Product	`log(xy) = log x + log y`	Multiply two numbers → ADD their logs.
Quotient	`log(x/y) = log x − log y`	Divide → SUBTRACT logs.
Reciprocal	`log(1/x) = − log x`	Reciprocal of x flips the sign of log x.
Negative power	`log(x⁻ⁿ) = −n · log x`	Negative exponent flips the sign and pulls the coefficient out.
General product chain	`log(x₁ x₂ x₃ …) = Σ log xᵢ`	Multiplying any number of factors → sum the logs.

🎈 Plain English

Multiplication is hard, addition is easy. Logs turn multiplications INTO additions and divisions into subtractions. That's why slide-rules used to be the world's calculators before computers existed.

📏 4 · Power & Root Rules

Rule	Formula	Plain meaning
Power	`log(xⁿ) = n · log x`	Pull the power out as a multiplier — the single most-used log rule in NEET kinetics.
Root	`log(ⁿ√x) = (1/n) · log x`	Same as power rule with fractional exponent 1/n.
Square root	`log √x = (1/2) log x`	Special case of the root rule.
Power in base	`log_aⁿ x = (1/n) · log_a x`	Raise the BASE to a power → divide the log by n.
Power on both sides	`log_aⁿ x^m = (m/n) · log_a x`	Combine the two above.

📌 Example

log(8³) = 3 · log 8 = 3 · 3 log 2 = 9 × 0.30 = 2.71

📌 Example (power in base)

log₉ 27 = log_3²(3³) = (3/2) · log₃ 3 = 3/2

4.1 Nested / Tower Powers — log(x^{y^z}) and log((x^y)^z)

These two look similar but they are different. The standard convention for exponentiation is right-associative, so x^{y^z} means x^{(y^z)}, not (x^y)^z.

Expression	What it means	Apply log → answer
`log(x^{y^z})`	`log(x^{(y^z)})` (tower)	`y^z · log x`
`log((x^y)^z)`	`log(x^yz)` (rule of indices)	`yz · log x`
`log(x^y · x^z)`	`log(x^y+z)`	`(y + z) · log x`
`log(x^y/x^z)`	`log(x^y−z)`	`(y − z) · log x`

🎈 The rule in one line

The power rule log(xⁿ) = n · log x works no matter what shape n has. n can be a number (3), a variable (y), a product (yz), a sum (y + z), a power (y^z), or even another logarithm. Whatever sits in the exponent comes out as the multiplier — full stop.

4.2 Step-by-step worked examples

📌 Example 1 — pure tower

log(2^3²). The exponent is 3² = 9. So log(2⁹) = 9 · log 2 = 9 × 0.30 = 2.70.

Compare with log((2³)²) = log(2⁶) = 6 · log 2 = 6 × 0.30 = 1.80. Notice the two are completely different.

📌 Example 2 — symbolic tower

Simplify log(x^{y^z}).

Apply the power rule with n = y^z: → y^z · log x.

If you also want to crack y^z, take a second log: log(y^z · log x) = z · log y + log(log x). (You can keep peeling layers off.)

📌 Example 3 — variable in BOTH base & tower

Simplify log(x^{x^x}).

The exponent is x^x. → x^x · log x. (Then if you take log of the result you can split x^x further as x · log x + log(log x).)

📌 Example 4 — bracketed power (different!)

Simplify log((x²)³).

Brackets force the rule of indices first: (x²)³ = x^2·3 = x⁶.

→ log(x⁶) = 6 · log x.

📌 Example 5 — JEE-style logarithmic differentiation setup

Differentiate y = x^{(sin x)}.

Take log: log y = sin x · log x (the power sin x comes out front).

Differentiate both sides: (1/y) · dy/dx = cos x · log x + sin x · (1/x).

So dy/dx = x^{(sin x)}·[cos x · log x + sin x/x].

⚡ Common NEET / JEE trap: students write log(x^{y^z}) = yz · log x. That's wrong. yz is what you get from (x^y)^z only. The pure tower without brackets gives y^z·log x.

4.3 What about log_x^y(x^z)?

Use the "power in BOTH" rule (4.2 of section 4 main table):

log_x^y(x^z) = (z/y) · log_x x = z/y

📌 Example

log₄ 32 = log_2² 2⁵ = 5/2 = 2.5.

📏 5 · Change of Base & Base Swapping

Rule	Formula	Plain meaning
Change of base	`log_b x = log_a x / log_a b`	Convert from any base to any other — divide by the log of the new base.
Swap base ↔ argument	`log_a b = 1 / log_b a`	Swap the base and the argument → take the reciprocal.
Product of swaps	`log_a b · log_b a = 1`	Direct consequence of swap rule.
Three-base chain	`log_a b · log_b c = log_a c`	"Chain rule" — the middle base cancels.
n-base chain	`log_a₁ a₂ · log_a₂ a₃ · … · log_aₙ₋₁ aₙ = log_a₁ aₙ`	Generalisation of three-base chain.

📌 Example (change of base)

log₂ 32 = log 32 / log 2 = (5 log 2) / log 2 = 5 (since 2⁵ = 32)

📏 6 · Natural Log ↔ Common Log Conversion

Rule	Formula	Plain meaning
ln to log	`ln x = 2.303 · log₁₀ x`	The single most-used conversion in chemistry kinetics — multiply log by 2.303.
log to ln	`log₁₀ x = 0.4343 · ln x`	= ln x / 2.303 = ln x · log e.
ln of e^x	`ln(e^x) = x`	Natural log and exponential cancel.
e to the ln	`e^{ln x} = x`	The other direction also cancels.
10 to the log	`10^{log x} = x`	"Antilog of log of x is x."
ln 1 / ln e / ln 10	`ln 1 = 0, ln e = 1, ln 10 = 2.303`	Anchors.

⚡ 2.303 is literally ln 10. Memorise it once — it appears in Arrhenius, Nernst, radioactive decay, ΔG° = −RT ln K conversion to −2.303 RT log K, and the integrated first-order rate law.

📏 7 · Sign & Domain Rules

Situation	Value of log₁₀ x	Why
x > 1	Positive	x is bigger than 10⁰ = 1.
x = 1	0	Any base to power 0 = 1.
0 < x < 1	Negative	Need a negative power of 10 to shrink past 1.
x = 0	Undefined (→ −∞)	No finite power makes a positive base equal 0.
x < 0	Undefined (in real numbers)	No real exponent of a positive base gives a negative.
Base a ≤ 0 or a = 1	Not allowed	Base must be positive and ≠ 1.

🎈 Plain English

Logs only know how to handle positive arguments. Zero is the cliff edge (log → −∞), negatives fall off the cliff entirely (no real value).

📏 8 · Inequality (Monotonicity) Rules

If…	Then…
Base `a > 1` & `x > y > 0`	`log_a x > log_a y` (log is increasing)
Base `0 < a < 1` & `x > y > 0`	`log_a x < log_a y` (log is decreasing — flips!)
For any base, x = y	`log x = log y`
x = y means base & argument identical	log = 1

🎈 Plain English

For "normal" bases (10 or e), the log function is increasing: bigger input → bigger log. NEET / JEE always uses base > 1, so the second case rarely shows up.

📏 9 · Useful Derived Results

Rule	Formula
Sum of log → log of product	`log a + log b + log c = log(abc)`
Coefficient of log → log of power	`n · log a = log(aⁿ)`
Log of a quotient power	`log((a/b)ⁿ) = n · (log a − log b)`
Log of a square root quotient	`log √(a/b) = ½(log a − log b)`
Log of factorial (large n)	`log 1 + log 2 + log 3 + … + log n = log(n!)`
Log of m · √n	`log(m·√n) = log m + ½ log n`
Log of e^kx	`ln(e^kx) = kx`
Log of a · 10ⁿ	`log(a × 10ⁿ) = log a + n` (scientific notation)

⚡ The scientific-notation rule is gold for pH calculations: log(3.16 × 10⁻⁵) = log 3.16 − 5 ≈ 0.5 − 5 = −4.5 → pH = 4.5.

∑ 10 · Log Values You Must Memorise

Memorise three numbers (log 2, log 3, log 7) and you can compute almost every log NEET / JEE will ask.

Number	log₁₀ value	Derivation
1	0	10⁰ = 1
2	0.3010	memorise
3	0.4771	memorise
4	0.6020	= 2 × log 2
5	0.6990	= 1 − log 2 (since 5 = 10/2)
6	0.7781	= log 2 + log 3
7	0.8451	memorise
8	0.9030	= 3 × log 2
9	0.9542	= 2 × log 3
10	1	10¹ = 10

10.1 Natural log values

Constant	Value
`ln 2`	0.693
`ln 3`	1.099
`ln 5`	1.609
`ln 10`	2.303
`ln e`	1
`e`	2.718

10.2 Antilog mental estimates

If log x ≈	then x ≈
0	1
0.30	2
0.48	3
0.60	4
0.70	5
0.78	6
0.90	8
0.95	9
1	10
2	100
3	1000

⚡ Characteristic and mantissa. Any log has two parts. log 250 = 2.398: the 2 (characteristic) comes from "digits before decimal − 1" — 250 has 3 digits → 2. The 0.398 (mantissa) is log 2.5.

🧪 11 · Chemistry Applications of Logarithms

Every NEET Chemistry chapter that involves "very large" or "very small" numbers uses logs.

11.1 Ionic Equilibrium

pH = − log[H⁺] pOH = − log[OH⁻] pK_a = − log K_a pK_b = − log K_b

pH + pOH = 14 (at 25 °C) pK_a + pK_b = 14

Henderson (acidic buffer): pH = pK_a + log ([salt]/[acid])

Weak acid: pH = ½ (pK_a − log C)

11.2 Chemical Kinetics

First-order: k = (2.303/t) · log([A]₀/[A])

Half-life (first-order): t½ = 0.693/k = ln 2 / k

Arrhenius: log(k₂/k₁) = (E_a/2.303 R) · (1/T₁ − 1/T₂)   |   ln(k₂/k₁) = (E_a/R) · (1/T₁ − 1/T₂)

11.3 Electrochemistry — Nernst

E = E° − (0.0591/n) · log Q (at 298 K)

E° = (0.0591/n) · log K (at equilibrium)

For H₂ electrode: E = − 0.0591 · pH

11.4 Thermodynamics

ΔG° = − RT ln K = − 2.303 RT log K

At 298 K: ΔG° (kJ/mol) ≈ − 5.7 · log K

11.5 Beer-Lambert Spectroscopy

A = log(I₀/I) = ε · b · c

🧪 NEET trap

Absorbance is base-10 log, not natural log. Transmittance T = I/I₀ ⇒ A = − log T.

⚛️ 12 · Physics Applications of Logarithms

12.1 Modern Physics — Radioactive decay

N(t) = N₀ · e^−λt ⇒ ln(N/N₀) = − λt

t = (2.303/λ) · log(N₀/N) t½ = ln 2 / λ = 0.693/λ

12.2 Sound — Intensity Level

β (dB) = 10 · log(I/I₀)

🎈 Plain English

The decibel scale is logarithmic because the ear handles intensities over a billion-fold range. +10 dB = 10× intensity; +20 dB = 100× intensity.

12.3 Optics & Spectroscopy

Absorbance A = log(I₀/I)

Magnitude scale (astronomy): m₁ − m₂ = − 2.5 log(F₁/F₂)

12.4 Misc physics derivations using logs

Capacitor RC charging: Q = Q₀(1 − e^−t/τ) ⇒ t = − τ · ln(1 − Q/Q₀)
Boltzmann distribution: ln(N₂/N₁) = − (E₂ − E₁)/(kT)
Entropy (statistical): S = k · ln W
Maxwell-Boltzmann ratio: ln(f(v₂)/f(v₁)) = − m(v₂² − v₁²)/2kT

📝 13 · Worked NEET / JEE Examples

📌 Example 1 · pH from [H⁺]

If [H⁺] = 3.16 × 10⁻⁵, then pH = − log(3.16 × 10⁻⁵) = 5 − log 3.16 ≈ 5 − 0.5 = 4.5.

📌 Example 2 · Weak acid pH

0.1 M acetic acid, K_a = 1.8 × 10⁻⁵. pK_a = − log(1.8 × 10⁻⁵) = 5 − log 1.8 ≈ 5 − 0.255 = 4.745. pH = ½(pK_a − log C) = ½(4.745 + 1) = 2.87.

📌 Example 3 · Arrhenius numerical

k doubles from 300 K to 310 K. Find E_a.
log 2 = (E_a/2.303 R)(1/300 − 1/310). log 2 = 0.301. RHS factor = (10/300×310) ≈ 1.075×10⁻⁴. E_a = (0.301 × 2.303 × 8.314)/(1.075×10⁻⁴) ≈ 53.6 kJ/mol.

📌 Example 4 · Nernst

Cell with E° = 0.59 V and n = 1. log K = E°·n/0.0591 = 0.59/0.0591 = 10 → K = 10¹⁰.

📌 Example 5 · First-order half-life

k = 5 × 10⁻⁴ s⁻¹. t½ = 0.693/k = 0.693/(5 × 10⁻⁴) = 1386 s.

📌 Example 6 · ΔG° from K

K = 10⁵ at 298 K. ΔG° = − 2.303 RT log K = − 2.303 × 8.314 × 298 × 5 = − 28.5 kJ/mol. Shortcut: ΔG° (kJ) ≈ − 5.7 × log K = − 5.7 × 5 = −28.5 kJ/mol.

📌 Example 7 · Radioactive decay

Half-life 10 years. After 40 years, fraction remaining = (1/2)⁴ = 1/16. Equivalent log form: ln(N/N₀) = − λt = − 4 ln 2 → N/N₀ = e^{−4 ln 2} = 2⁻⁴ = 1/16. 1/16 of N₀.

📌 Example 8 · Buffer Henderson

Acetic acid + sodium acetate, both 0.1 M. pH = pK_a + log(salt/acid) = pK_a + log 1 = pK_a = 4.74.

📌 Example 9 · Absorbance

If transmittance T = 0.1, then A = − log 0.1 = 1. If T = 0.01, A = 2. Each unit of A = 10× drop in transmitted light.

📌 Example 10 · Decibel

Doubling sound intensity adds 10 log 2 ≈ 3 dB. Ten-fold increase adds 10 log 10 = 10 dB. Hundred-fold adds 20 dB.

🔑 14 · One-Page Cheat Sheet

The Three Rules NEET Tests in 90% of Questions

Rank	Rule	Most common use
1	Product / Power `log(xy) = log x + log y`, `log(xⁿ) = n log x`	pH, K calculations, kinetics
2	ln ↔ log `ln x = 2.303 log x`	Arrhenius, Nernst, radioactive decay
3	Inverse identity `10^{log x} = x`, `e^{ln x} = x`	antilog, finding [H⁺] from pH, finding N from ln(N/N₀)

Anchor values

Constant	Value
log 2 / log 3 / log 7	0.30 / 0.48 / 0.85
ln 2 / ln 10	0.693 / 2.303
0.0591 V	= (2.303 RT/F) at 298 K — Nernst factor
0.693/k	First-order half-life formula
5.7 kJ/mol	= 2.303 RT at 298 K → ΔG° (kJ) ≈ −5.7 log K

Sign quick check

x > 1 → log x > 0
x = 1 → log x = 0
0 < x < 1 → log x < 0
x ≤ 0 → undefined

⚡ Read this card daily for a week before the exam. Page-1 Chemistry / Physics questions reward this single page more than 100 pages of mock tests.

📏 1 · The Basics — what a logarithm is

1.1 Three parts of a log expression

1.2 The two bases NEET cares about

📏 2 · Basic Identities

📏 3 · The Three Operation Rules (most useful!)

📏 4 · Power & Root Rules

4.1 Nested / Tower Powers — log(xyz) and log((xy)z)

4.2 Step-by-step worked examples

4.3 What about logxy(xz)?

📏 5 · Change of Base & Base Swapping

📏 6 · Natural Log ↔ Common Log Conversion

📏 7 · Sign & Domain Rules

📏 8 · Inequality (Monotonicity) Rules

📏 9 · Useful Derived Results

∑ 10 · Log Values You Must Memorise

10.1 Natural log values

10.2 Antilog mental estimates

🧪 11 · Chemistry Applications of Logarithms

11.1 Ionic Equilibrium

11.2 Chemical Kinetics

11.3 Electrochemistry — Nernst

11.4 Thermodynamics

11.5 Beer-Lambert Spectroscopy

⚛️ 12 · Physics Applications of Logarithms

12.1 Modern Physics — Radioactive decay

12.2 Sound — Intensity Level

12.3 Optics & Spectroscopy

12.4 Misc physics derivations using logs

📝 13 · Worked NEET / JEE Examples

🔑 14 · One-Page Cheat Sheet

The Three Rules NEET Tests in 90% of Questions

Anchor values

Sign quick check

4.1 Nested / Tower Powers — log(x^{y^z}) and log((x^y)^z)

4.3 What about log_x^y(x^z)?