Logarithms

Logarithms give NDA 1–3 direct questions and also appear inside calculus and indices problems. The chapter is small, entirely formula-based, and among the fastest to score in Paper I.

Core ideas

By definition, log_a N = x means aˣ = N, with a > 0, a ≠ 1, N > 0. Logs of zero or negatives are undefined.

Laws:

• log(mn) = log m + log n
• log(m/n) = log m − log n
• log(mⁿ) = n log m
• Change of base: log_a N = log_b N / log_b a
• log_a a = 1, log_a 1 = 0

Useful: a^(log_a N) = N; log_a b × log_b a = 1; log_(aⁿ) (mᵏ) = (k/n) log_a m.

Common vs natural: log usually means base 10 (common); ln means base e (natural). log 2 ≈ 0.3010, log 3 ≈ 0.4771 — handy for characteristic/mantissa items.

Characteristic and mantissa: log of a number = (integer characteristic) + (non-negative mantissa). Number of digits in N = characteristic of log₁₀N, plus 1.

Exam Tricks & Tips

• 🎯 'Number of digits in aᵏ' = ⌊k·log₁₀a⌋ + 1 — a classic NDA question solved with log 2 and log 3.
• 🎯 Convert an equation like log_a x = b straight to x = aᵇ; don't manipulate logs unnecessarily.
• 🎯 log_a b = 1/log_b a — flip the base to simplify chained logs.
• 🎯 Bring exponents down with log(mⁿ) = n log m before doing anything else.
• 🎯 If the base is unstated in NDA numerical work, assume base 10.
• ❌ Common mistake: writing log(m+n) = log m + log n. The product rule is for multiplication only; there is NO simple rule for a sum inside a log.

Expected exam pattern

Expect: simplify a log expression, solve a simple log equation, count digits in a large power, or use change of base. Values of log 2 and log 3 are often provided or assumed.

Quick recap

Remember log(mn)=log m+log n, log(m/n)=log m−log n, log mⁿ=n log m, change of base, and log_a1=0, log_aa=1. Digits in aᵏ = ⌊k log₁₀a⌋+1. Never split a log of a sum.

Cover, recall, check. 10 cards on log laws.

Q1. Definition: log_a N = x means?
A1. aˣ = N, with a>0, a≠1, N>0.

Q2. log(mn) = ?
A2. log m + log n.

Q3. log(m/n) = ?
A3. log m − log n.

Q4. log(mⁿ) = ?
A4. n log m.

Q5. Change of base formula?
A5. log_a N = log_b N / log_b a.

Q6. log_a 1 and log_a a?
A6. log_a 1 = 0; log_a a = 1.

Q7. Value of a^(log_a N)?
A7. N.

Q8. log_a b × log_b a = ?
A8. 1 (they are reciprocals).

Q9. Number of digits in a positive integer N?
A9. ⌊log₁₀ N⌋ + 1.

Q10. Is log(m+n) = log m + log n valid?
A10. No — there is no log rule for a sum; the product rule needs multiplication.

Logarithms turn multiplication into addition and powers into products — the reason they underpin so many quick NDA calculations and appear as standalone MCQs.

Core idea / what this tests: The definition log_a x = y ⇔ aʸ = x, the three algebraic laws, base change, and using them to compare or simplify expressions.

Deep explanation

Beginner — Definition and basics

log_a x = y means aʸ = x, valid for a > 0, a ≠ 1, x > 0. Immediate consequences:

• log_a 1 = 0 (since a⁰ = 1).
• log_a a = 1.
• a^(log_a x) = x.
  Common log uses base 10 (written log), natural log uses base e (written ln).

Intermediate — The three laws

• Product: log(xy) = log x + log y.
• Quotient: log(x/y) = log x − log y.
• Power: log(xᵖ) = p·log x.
  These convert a complicated product/power into a simple sum, which is exactly why log tables once ruled computation.

Advanced — Change of base and comparisons

Change of base: log_a x = log_b x / log_b a = 1 / log_x a.
So log_a b · log_b a = 1, and chains telescope: log_a b · log_b c · log_c d = log_a d.
To compare logs with different bases, bring them to a common base first. Remember: log is increasing for base > 1, so larger x gives larger log.

Worked example

Evaluate log₂ 8 + log₃ (1/9).
log₂ 8 = log₂ 2³ = 3. log₃ (1/9) = log₃ 3⁻² = −2.
Sum = 3 + (−2) = 1.

NDA relevance

Direct evaluation (like the example), simplifying nested logs, solving log equations, and "which is greater" comparisons all appear. Logarithms also shorten arithmetic in physics-flavoured GAT problems and in number-of-digits questions (digits of N = ⌊log₁₀ N⌋ + 1).

Exam tricks & shortcuts

• Express arguments as powers of the base to read the log off instantly (log₂ 32 = 5).
• Number of digits in N = ⌊log₁₀ N⌋ + 1 — a classic one-liner.
• log_a b = 1/log_b a — flip the base to simplify awkward fractions.
• Mnemonic: "multiply→add, divide→subtract, power→pull down".

Key formulas

• Definition: log_a N = x ⇔ aˣ = N (a>0, a≠1, N>0).
• log(mn) = log m + log n; log(m/n) = log m − log n; log mᵖ = p log m.
• Change of base: log_a N = log_b N/log_b a; log_a b = 1/log_b a.
• log_a a = 1; log_a 1 = 0; a^{log_a N} = N.
• Common log base 10, natural log base e.
• Characteristic + mantissa for computation.

Usage: convert products/powers to sums using log laws before computing.

Worked Example

Problem: Given log₁₀ 2 = 0.3010, find the number of digits in 2⁶⁴.

Solution:
The number of digits in a positive integer N is floor(log₁₀ N) + 1.

Compute the logarithm:
log₁₀(2⁶⁴) = 64 × log₁₀ 2 = 64 × 0.3010 = 19.264.

This tells us 2⁶⁴ lies between 10^19 and 10^20 (since 19 < 19.264 < 20). Taking the integer part:
floor(19.264) = 19.

Therefore the number of digits = 19 + 1 = 20.

Answer: 2⁶⁴ has 20 digits. (Indeed 2⁶⁴ = 18,446,744,073,709,551,616, which has 20 digits.)

Solution note: The rule works because log₁₀ N = (integer part) + (fractional part); the integer part fixes the power of 10, and adding 1 accounts for the leading digit.