NEET Physics — Fundamental Mathematics and Vector

Logarithm – Complete Notes, Revision, Important Questions & Downloads

Logarithm in NEET Physics covers two subtopics: Definition and types of logarithm, and Important formulae of logarithm. In physics, logarithms appear whenever exponential relationships arise — radioactive decay uses ln(N/N₀) = −λt to find half-life, sound intensity is measured via β = 10 log₁₀(I/I₀) decibels, and RC circuit discharge follows V = V₀ e^{−t/RC} requiring natural log for time extraction. NEET rarely tests logarithms directly but expects fluent use of properties like log_a(mn) = log_a m + log_a n and the conversion log_e x = 2.3026 log₁₀ x when solving physics numericals. For instance, computing the time for a radioactive sample to decay to 1/8 of its initial value requires recognizing that log₂ 8 = 3, giving t = 3T₁/₂.

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2 SubtopicsPrerequisite MathUsed in Decay & Sound

Expected QuestionsQ

0–1

Direct logarithm questions are extremely rare in NEET; however, logarithmic manipulation is embedded in radioactive decay, sound intensity (decibel), and RC/LR circuit time-constant problems.

Time Required⏱

1–2 hours

Brief study of definitions plus the four core properties, followed by physics-context practice applying logarithms in decay and sound problems.

Difficulty⚡

Easy

The formulae are straightforward; the challenge lies in recognising when to apply logarithms inside a physics equation involving exponential relationships.

NRI USA Curriculum GapUS

Low–Medium

US Pre-Calculus covers logarithmic properties thoroughly, but students may not have practised applying log_e to log₁₀ conversion in physics contexts like decibel calculations or half-life computations.

2Subtopics

8+Practice Questions

4Free Downloads

1–2 hrsPrep Time

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NEET Weightage — Logarithm

Fundamental Mathematics and Vector (Chapter 0)

NEET Year	Questions from this Topic	Bar	Marks
2024	0	0 Q	0
2023	0	0 Q	0
2022	0	0 Q	0
2021	1	1 Q	4
2020	0	0 Q	0
2019	0	0 Q	0
6-Year Total (2019–2024)	0–2		0–8

The conversion factor log_e x = 2.3026 log₁₀ x appears in radioactive decay calculations — NEET expects you to switch between natural and common logarithms when solving for half-life or decay constant.
The product and quotient rules log_a(mn) = log_a m + log_a n and log_a(m/n) = log_a m − log_a n simplify intensity-ratio problems in sound (decibel scale β = 10 log₁₀(I/I₀)) and light (optical density).
The power formula log_a m^n = n log_a m is the key step when solving N = N₀(1/2)^{t/T₁/₂} for time t — take log of both sides and use log(1/2) = −log 2 to isolate the exponent.

📊

~0–0.5

Avg Questions / Year

🎯

0–8

Total Marks (6 yrs)

📐

Indirect

Pattern

⚡

Easy

Difficulty

Exam Strategy for Logarithm in NEET Physics

Memorise the four core logarithm properties cold Commit log_a(mn) = log_a m + log_a n (product), log_a(m/n) = log_a m − log_a n (quotient), log_a m^n = n log_a m (power), and log_a m = log_b m × log_a b (base change) to instant recall. Before applying any property, confirm you have identified the correct base. The trap: confusing log_a(m+n) with log_a m + log_a n — unlike multiplication, addition inside a logarithm has no simplification rule.

Drill the natural-to-common log conversion Memorise log_e x = 2.3026 log₁₀ x (equivalently, log₁₀ e ≈ 0.4343). When a NEET problem gives ln 2 = 0.693, you can convert: log₁₀ 2 = 0.693/2.3026 ≈ 0.301. The trap: substituting 2.3026 when the problem already provides the natural log value directly, doubling the conversion.

Apply logarithms to isolate exponents in decay and circuit problems When you see N = N₀ e^{−λt} or V = V₀ e^{−t/RC}, take natural log of both sides: ln(N/N₀) = −λt, then solve for t. Recognise that log₂(8) = 3 means three half-lives. The trap: forgetting the negative sign in the exponent, which reverses the direction of the inequality when solving for time.

Download Study Notes — Logarithm

PDF · Cheat Sheet · MCQ Set · PYQ

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Logarithm — Full Notes

Complete notes covering the definition of logarithm with base concepts, common vs natural logarithm distinction, all four key properties (product, quotient, power, base change), and worked physics examples in radioactive decay and sound intensity.

2 subtopicsWorked examplesPhysics context

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Logarithm — Formula Sheet

One-page formula reference: log definition, four core properties, log_e to log₁₀ conversion factor 2.3026, and antilogarithm — key formulas, conditions, and one worked example per subtopic.

1 pageAll key formulas

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Logarithm — MCQ Practice

10 NEET-style MCQs testing logarithmic manipulation in physics contexts: half-life computation using ln 2, decibel level calculation with log₁₀, and RC time-constant extraction using natural logarithm.

10 MCQsDetailed solutions

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Logarithm — NEET-Style PYQ Practice

Collection of NEET-style practice questions where logarithmic properties determine the answer — decay constant from half-life data, sound intensity ratios converted to dB, and exponential discharge timing in RC circuits.

NEET-styleAnswer key included

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Subtopics in Logarithm

2-Column Table

Column A	Column B
Definition and types of logarithm↗	Important formulae of logarithm↗

Rapid Revision — Logarithm

Concept → Trap → Example

1) Definition and types of logarithm

Foundation Concept

If a^x = N then log_a N = x. Common logarithm: base 10 (log₁₀). Natural logarithm: base e (log_e or ln). Conversion: log_e x = 2.3026 log₁₀ x.

The base must be positive and not equal to 1 — log₁ N is undefined because 1 raised to any power is always 1, never yielding N ≠ 1.
Natural logarithm (base e ≈ 2.718) is the default in physics equations involving exponential decay or growth; common logarithm (base 10) is used in decibel and pH calculations.
Common NEET trap: confusing log₁₀ and log_e values — ln 2 = 0.693 but log₁₀ 2 = 0.301. Using one in place of the other changes the answer by a factor of 2.3026.

Example (NEET-style)Convert ln 2 = 0.693 to common log: log₁₀ 2 = 0.693 / 2.3026 = 0.301. Verify: 10^{0.301} = 2.00 (approximately). In NEET, if a problem gives λ = 0.693/T₁/₂, recognise this is ln 2 / T₁/₂.

2) Important formulae of logarithm

Property Toolkit

Product: log_a(mn) = log_a m + log_a n. Quotient: log_a(m/n) = log_a m − log_a n. Power: log_a m^n = n log_a m. Base change: log_a m = log_b m × log_a b.

The product rule converts multiplication inside the log into addition outside — useful when a NEET problem involves log of a product of physical quantities.
The power rule is the most frequently used in physics: to solve 2^n = 1024, take log of both sides to get n log 2 = log 1024, so n = log 1024 / log 2 = 10.
Common NEET trap: applying log_a(m + n) = log_a m + log_a n — this is false. The product rule applies only to log_a(m × n), not to log of a sum.

Example (NEET-style)Simplify log₁₀(8 × 125): log₁₀(8 × 125) = log₁₀ 8 + log₁₀ 125 = log₁₀ 2³ + log₁₀ 5³ = 3 log₁₀ 2 + 3 log₁₀ 5 = 3(log₁₀ 2 + log₁₀ 5) = 3 log₁₀ 10 = 3 × 1 = 3.

US Curriculum Gaps — Logarithm for NEET Physics

NRI students from US high schools may find these specific gaps when preparing for NEET Physics logarithm problems.

Natural-to-Common Logarithm Conversion in Physics (not practised in AP Physics 1 / AP Physics C)

US AP Physics courses provide calculators that handle log conversions automatically. NEET requires students to convert between log_e and log₁₀ by hand using the factor 2.3026, and to recognise ln 2 = 0.693 and log₁₀ 2 = 0.301 from memory during radioactive decay and sound intensity problems.

AP Physics 1 and AP Physics C do not test manual conversion between natural and common logarithms.
NEET expects students to use log_e x = 2.3026 log₁₀ x without a calculator to solve for decay time or intensity ratios.
Practice converting ln values to log₁₀ values by dividing by 2.3026, and memorise ln 2 = 0.693, ln 10 = 2.3026.

Logarithmic Properties Applied to Physics Equations (limited in US Pre-Calculus / Algebra 2)

US Pre-Calculus and Algebra 2 courses teach logarithm properties abstractly with pure-math examples. NEET embeds these properties inside physics equations — e.g., using the power rule to extract time from N = N₀(1/2)^{t/T₁/₂} or the quotient rule to simplify β = 10 log₁₀(I/I₀) when comparing two intensity levels.

US courses rarely require applying log properties to isolate variables in exponential physics equations.
NEET problems may phrase logarithm application as 'find the number of half-lives' or 'calculate the decibel difference', requiring identification of the log property from a physics context.
Practice rewriting physics exponentials in log form: if N/N₀ = e^{−λt}, then λt = −ln(N/N₀) = ln(N₀/N).

NEET-Style Practice Questions — Logarithm

4 NEET-style practice questions

1The half-life of a radioactive substance is 20 minutes. The time required for the substance to decay to 1/8 of its initial amount is:NEET-style practice

40 minutes

60 minutes

80 minutes

160 minutes

We need N/N₀ = 1/8. Since 1/8 = (1/2)³, we require 3 half-lives. Using logarithms: N = N₀(1/2)^{t/T₁/₂}. Taking log of both sides: log(1/8) = (t/T₁/₂) log(1/2). Since log(1/8) = log(2^{−3}) = −3 log 2 and log(1/2) = −log 2, we get −3 log 2 = (t/20)(−log 2), so t/20 = 3, giving t = 60 minutes. Option (a) gives only 2 half-lives (decay to 1/4, not 1/8). Option (c) gives 4 half-lives (decay to 1/16). Option (d) gives 8 half-lives (decay to 1/256). The key logarithmic step is recognising log₂ 8 = 3.

2If the intensity of a sound increases by a factor of 1000, the increase in sound level in decibels is: (Given: log₁₀ 10 = 1)NEET-style practice

10 dB

30 dB

100 dB

3 dB

Sound level difference Δβ = 10 log₁₀(I₂/I₁). Here I₂/I₁ = 1000 = 10³. So Δβ = 10 log₁₀(10³) = 10 × 3 log₁₀ 10 = 10 × 3 × 1 = 30 dB. This uses the power formula log_a m^n = n log_a m directly. Option (a) applies only for a 10-fold increase (10 log₁₀ 10 = 10). Option (c) confuses the intensity ratio with the dB value. Option (d) uses natural log instead of common log (ln 1000 ≈ 6.9, not 3). The decibel formula always uses log₁₀, never log_e.

3A capacitor discharges through a resistance. The voltage drops to 1/e of its initial value V₀ in time τ = RC. The time for the voltage to drop to V₀/e² is:NEET-style practice

2τ

τ/2

τ²

The discharge equation is V = V₀ e^{−t/RC} = V₀ e^{−t/τ}. Setting V = V₀/e²: V₀/e² = V₀ e^{−t/τ}. Dividing both sides by V₀: e^{−2} = e^{−t/τ}. Taking natural log of both sides: −2 = −t/τ, so t = 2τ. Here we used the property that ln(e^x) = x. Option (a) gives V₀/e, only one time constant of decay. Option (c) gives V₀ e^{−1/2} = V₀/√e, not V₀/e². Option (d) has incorrect dimensions (τ² has units of s², not s). The logarithmic step is taking ln of both sides to equate exponents.

4If log₁₀ 2 = 0.301, then the number of digits in 2¹⁰ is:NEET-style practice

The number of digits in a positive integer N is given by floor(log₁₀ N) + 1. Here log₁₀(2¹⁰) = 10 log₁₀ 2 = 10 × 0.301 = 3.01. So the number of digits = floor(3.01) + 1 = 3 + 1 = 4. Indeed, 2¹⁰ = 1024, which has 4 digits. This uses the power formula log_a m^n = n log_a m. Option (a) forgets the +1 in the digit-counting formula. Option (c) confuses the exponent 10 with the digit count. Option (d) takes only the integer part of log₁₀ 2. The power rule converts the problem from exponentiation to simple multiplication.

Practice Problems — Logarithm in Physics

Click "Reveal Answer" after attempting

1A radioactive sample has a decay constant λ = 0.0231 per minute. Using ln 2 = 0.693, find the half-life of the sample.

30 minutes

20 minutes

15 minutes

45 minutes

👁 Reveal Answer

Option (a): 30 minutes. The half-life formula is T₁/₂ = ln 2 / λ = 0.693 / 0.0231 = 30 minutes. This directly uses the definition: at t = T₁/₂, N = N₀/2, so ln(1/2) = −λT₁/₂, giving T₁/₂ = ln 2 / λ.

2Two sounds have intensities I₁ = 10⁻⁸ W/m² and I₂ = 10⁻⁵ W/m². The difference in their sound levels is:

3 dB

30 dB

300 dB

0.3 dB

👁 Reveal Answer

Option (b): 30 dB. Δβ = 10 log₁₀(I₂/I₁) = 10 log₁₀(10⁻⁵/10⁻⁸) = 10 log₁₀(10³) = 10 × 3 = 30 dB. The quotient rule simplifies the ratio inside the log: log(10⁻⁵) − log(10⁻⁸) = −5 − (−8) = 3.

3Simplify: log₂ 32 + log₂(1/4) − log₂ 2.

👁 Reveal Answer

Option (a): 2. log₂ 32 = log₂ 2⁵ = 5. log₂(1/4) = log₂ 2⁻² = −2. log₂ 2 = 1. Sum = 5 + (−2) − 1 = 2. Uses the power formula log_a m^n = n log_a m for each term, then simple addition.

4An RC circuit has R = 10 kΩ and C = 100 µF. How long does it take for the charge on the capacitor to drop to 10% of its initial value? (Given: ln 10 = 2.3026)

2.3026 s

1.0 s

0.4343 s

4.6052 s

👁 Reveal Answer

Option (a): 2.3026 s. τ = RC = 10×10³ × 100×10⁻⁶ = 1 s. Charge: Q = Q₀ e^{−t/τ}. Setting Q = 0.1 Q₀: 0.1 = e^{−t/1}. Taking ln: ln(0.1) = −t. Since ln(0.1) = −ln(10) = −2.3026, we get t = 2.3026 s. Uses the property ln(1/a) = −ln(a).

Physics — Logarithm Revision Checklist

Check off chapters as you revise

Use this section for quick chapter tracking before mocks, part tests, and final NEET revision.

Definition and types of logarithm: State the definition and identify the baseWrite the definition: if a^x = N then log_a N = x. Given 10³ = 1000, write log₁₀ 1000 = 3. Given e² ≈ 7.389, write ln 7.389 = 2.

Definition and types of logarithm: Convert between natural and common logarithmConvert ln 5 to log₁₀ 5 using log₁₀ 5 = ln 5 / 2.3026 = 1.6094 / 2.3026 ≈ 0.699. Verify: 10^{0.699} ≈ 5.

Definition and types of logarithm: Distinguish log_e and log₁₀ in a physics formulaIn the decay equation N = N₀ e^{−λt}, identify that taking ln (not log₁₀) of both sides gives ln(N/N₀) = −λt directly.

Important formulae of logarithm: Apply the product and quotient rulesExpand log₁₀(2 × 5) = log₁₀ 2 + log₁₀ 5 = 0.301 + 0.699 = 1.000. Then simplify log₁₀(50/2) = log₁₀ 50 − log₁₀ 2 = 1.699 − 0.301 = 1.398.

Important formulae of logarithm: Use the power and base-change rules to solve an exponent equationSolve 2^n = 1024: take log₁₀ of both sides to get n log₁₀ 2 = log₁₀ 1024. Using log₁₀ 2 = 0.301: n = 3.01/0.301 = 10. Verify: 2¹⁰ = 1024.

Frequently Asked Questions — Logarithm in NEET Physics

Notes · Downloads · Revision · Important Questions

Is Logarithm directly tested in NEET Physics?

Logarithm is almost never tested as a standalone question in NEET Physics. However, it is an essential mathematical tool embedded in topics like radioactive decay (half-life uses ln 2 = 0.693), sound intensity (decibel formula uses log₁₀), and capacitor discharge (time constant extraction uses natural log). Weakness in logarithm properties causes errors in these applied calculations.

What is the difference between common logarithm and natural logarithm?

Common logarithm uses base 10 and is written as log₁₀ N. Natural logarithm uses base e (≈ 2.71828) and is written as ln N or log_e N. In NEET Physics, natural logarithm appears in exponential decay/growth equations (N = N₀ e^{−λt}), while common logarithm appears in the decibel formula (β = 10 log₁₀(I/I₀)). The conversion between them is ln x = 2.3026 × log₁₀ x.

Why is ln 2 = 0.693 frequently given in NEET problems?

The value ln 2 = 0.693 is the natural logarithm of 2, and it appears directly in the radioactive decay formula: half-life T₁/₂ = ln 2 / λ = 0.693 / λ. Since NEET does not allow calculators, problems provide this value so you can compute half-life or decay constant numerically. If a problem gives λ = 0.0231/min, then T₁/₂ = 0.693/0.0231 = 30 min.

Can I apply the product rule to log(m + n)?

No. The product rule states log_a(m × n) = log_a m + log_a n — it applies only when the argument is a product, not a sum. There is no general simplification for log_a(m + n). This is one of the most common algebraic errors: students incorrectly write log(a + b) = log a + log b, which is false. Always check whether the argument involves multiplication or addition before applying any rule.

How do I use logarithms to find the number of half-lives in a decay problem?

If a sample decays to a fraction f of its original amount, then N/N₀ = (1/2)^n where n is the number of half-lives. Taking log of both sides: log(f) = n × log(1/2) = −n log 2. So n = −log(f)/log 2 = log(1/f)/log 2. For example, if f = 1/8: n = log(8)/log(2) = 3 log 2 / log 2 = 3 half-lives.

What is antilogarithm and when is it used?

Antilogarithm is the reverse of logarithm: if log N = x, then N = antilog(x) = 10^x (for common log) or N = e^x (for natural log). In physics, antilogarithm is used when you need to recover the original quantity after a logarithmic calculation. For example, if log₁₀ I = −5, then I = 10⁻⁵ W/m². The textbook defines it as: the number whose logarithm is x is called antilogarithm of x.

How does the base change formula help in NEET calculations?

The base change formula log_a m = log_b m × log_a b (equivalently, log_a m = log_b m / log_b a) lets you convert a logarithm from one base to another. In NEET, this is useful when a problem provides log₁₀ values but the equation uses ln (or vice versa). For instance, to find log₂ 8 using common logs: log₂ 8 = log₁₀ 8 / log₁₀ 2 = 0.903/0.301 = 3.

Why does the decibel formula use log₁₀ instead of ln?

The decibel scale β = 10 log₁₀(I/I₀) uses common logarithm (base 10) because the scale is designed so that each 10 dB increase corresponds to a 10-fold increase in intensity. Using ln instead would produce non-integer dB values for powers of 10, destroying the clean factor-of-10 relationship. When NEET states the decibel formula, always use log₁₀, not ln.

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Logarithm

Logarithm – Complete Notes, Revision, Important Questions & Downloads

1) Definition and types of logarithm

2) Important formulae of logarithm

Natural-to-Common Logarithm Conversion in Physics (not practised in AP Physics 1 / AP Physics C)

Logarithmic Properties Applied to Physics Equations (limited in US Pre-Calculus / Algebra 2)

Physics — Logarithm Revision Checklist

NEET NRI Counseling & Admission eBook Download

Schedule Trial Session For NEET Prep

Definition and types of logarithm

Important formulae of logarithm

Subtopics

Definition and types of logarithm

Important formulae of logarithm

NEET > Physics > Physical World and Measurement Chapters

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Logarithm

Overview content

Logarithm – Complete Notes, Revision, Important Questions & Downloads

1) Definition and types of logarithm

2) Important formulae of logarithm

Natural-to-Common Logarithm Conversion in Physics (not practised in AP Physics 1 / AP Physics C)

Logarithmic Properties Applied to Physics Equations (limited in US Pre-Calculus / Algebra 2)

Physics — Logarithm Revision Checklist

NEET NRI Counseling & Admission eBook Download

Schedule Trial Session For NEET Prep

Definition and types of logarithm

Important formulae of logarithm

Subtopics

Definition and types of logarithm

Important formulae of logarithm

NEET > Physics > Physical World and Measurement Chapters

Comments

Leave a comment

Comments (0)