NDA Maths · Teaching notes

Logarithms — NDA Maths

Logarithms is a small but reliable scorer: 27 PYQs span 2017–2026, mostly EASY/MODERATE, with a handful of HARD that hinge on one clever identity rather than heavy algebra. Almost every question reduces to a tiny toolkit — the three laws (product, quotient, power), the change-of-base rule and its reciprocal twin, and the discipline of checking the domain. The notes teach in two movements, foundations first: (1) Identities, Change of Base & Sums — what a logarithm IS, the laws that split and combine logs, the change-of-base rule that powers the recurring 1/log_k N telescoping sums, the sign and minimum-value questions, and logs sitting inside an AP/GP; (2) Solving Logarithmic Equations & Applications — taking the log of an exponential equation, the substitution t = aˣ that turns a log equation into a quadratic, the domain checks that decide how many solutions survive, the GP / chain-rule / AM-GM 'can never equal' conditions, and the trailing-zeros application. The single highest-yield idea is change of base — internalise log_b a = (log a)/(log b) and its consequence 1/log_a b = log_b a, and a third of the chapter becomes one-liners. Every PYQ is tagged.

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Subtopic notes

PYQ weightage by concept

10 concepts · 27 PYQs — where the marks actually sit, so you know what to drill first

Logarithm Identities, Change of Base & Sums16 PYQs · 59%

Concept	PYQs	Share
Applying the Laws — Evaluate and Combine	6	22%
Change of Base & the Reciprocal Identity	6	22%
Sign of a Logarithm & Bounds of a Log Function	2	7%
Logarithms in AP/GP and the Geometric Mean	2	7%
What a Logarithm Is — Laws, Special Values, Domainfoundation	—	—

Solving Logarithmic Equations & Applications11 PYQs · 41%

Concept	PYQs	Share
Taking the Log of an Exponential Equation	3	11%
Substitution t = aˣ to a Quadratic	3	11%
Domain Checks & Counting Solutions	2	7%
GP, Chain-Rule & AM-GM Conditions	2	7%
Application — Trailing Zeros of a Factorial	1	4%

Formula & revision sheet

10 formulas · 10 gotchas across all subtopics — the exam-eve cheat-sheet

Logarithm Identities, Change of Base & Sums

Formulas (5)

What a Logarithm Is — Laws, Special Values, Domain · The defining equivalence and the three laws $\log_a N = x \iff a^x = N;\quad \log_a(MN)=\log_a M+\log_a N,\ \ \log_a M^k = k\log_a M$
Applying the Laws — Evaluate and Combine · Rewrite as powers, then pull the exponent out $\log_a(b^m \cdot c^n) = m\log_a b + n\log_a c$
Change of Base & the Reciprocal Identity · Change of base and its reciprocal twin $\log_b a = \dfrac{\log a}{\log b}, \qquad \dfrac{1}{\log_a b} = \log_b a$
Sign of a Logarithm & Bounds of a Log Function · Sign of a log (base > 1) $\log_a N \;\begin{cases}>0 & N>1\\ =0 & N=1\\ <0 & 0<N<1\end{cases}$
Logarithms in AP/GP and the Geometric Mean · AP and GP conditions for three terms $\text{AP}: 2q = p+r, \qquad \text{GP}: q^2 = pr$

Watch out for (5)

$\log(M+N)$ is NOT $\log M + \log N$
→ What a Logarithm Is — Laws, Special Values, Domain
Keep the base when you pull out a power
→ Applying the Laws — Evaluate and Combine
Reciprocal flips the base and the argument together
→ Change of Base & the Reciprocal Identity
The minimum is of the LOG, not the quadratic
→ Sign of a Logarithm & Bounds of a Log Function
AP holding does not make it GP — test GP separately
→ Logarithms in AP/GP and the Geometric Mean

Solving Logarithmic Equations & Applications

Formulas (5)

Taking the Log of an Exponential Equation · Bring the exponent down with a log $a^x = b \;\Rightarrow\; x = \dfrac{\log b}{\log a}$
Substitution t = aˣ to a Quadratic · Let t = aˣ and solve the quadratic (keep t > 0) $t = a^x > 0,\quad t^2 + bt + c = 0 \;\Rightarrow\; x = \log_a t$
Domain Checks & Counting Solutions · Keep only roots with every argument > 0 $\log_a M = \log_a N \Rightarrow M = N,\ \text{then require } M,N > 0$
GP, Chain-Rule & AM-GM Conditions · Chain rule and the AM-GM floor $\log_x a\cdot\log_b x = \log_b a, \qquad t + \tfrac{1}{t} \ge 2\ (t>0)$
Application — Trailing Zeros of a Factorial · Legendre — trailing zeros count the 5s $Z(n) = \sum_{i\ge 1}\left\lfloor \dfrac{n}{5^i} \right\rfloor$

Watch out for (5)

$\log_{10} 0.2 = \log_{10} 2 - 1$ , which is negative
→ Taking the Log of an Exponential Equation
Throw out the non-positive t
→ Substitution t = aˣ to a Quadratic
An algebraic root is not a solution until the domain clears it
→ Domain Checks & Counting Solutions
The GP condition squares the MIDDLE term
→ GP, Chain-Rule & AM-GM Conditions
Count the 5s, not the 2s — and don't forget 25, 125…
→ Application — Trailing Zeros of a Factorial