NDA Mathematics · Formula sheet

Logarithms formulas

10 formulas and 13 common traps for NDA Mathematics Logarithms, grouped by subtopic.

Logarithm Identities, Change of Base & Sums

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

Common traps

$\log(M+N)$ is NOT $\log M + \log N$

The product law splits a log of a product, not a log of a sum.

\log_a(M+N)

has no simplification — only

\log_a(MN)

splits into

\log_a M+\log_a N

The base must be $>0$ and $\neq 1$

\log_a N

is only defined for a base

a > 0,\ a \neq 1

. A base of

1

is forbidden because

1^x = 1

for every

x

, so

\log_1 N

has no unique value; a base

\le 0

breaks the exponential entirely.

Keep the base when you pull out a power

\log_7\sqrt{7\sqrt{7\sqrt{7}}}

first becomes

\tfrac{7}{8}

, but a second outer

\log_7

of that gives

\log_7\tfrac{7}{8} = 1 - 3\log_7 2

— don't lose the

-\log_7 8 = -3\log_7 2

term.

Reciprocal flips the base and the argument together

\dfrac{1}{\log_a b} = \log_b a

— the base and argument swap. It does NOT equal

\log_a(1/b)

(which would be

-\log_a b

). Keep the two operations separate.

Change of base does NOT invert the fraction

\log_b a = \dfrac{\log a}{\log b}

— the NEW argument

a

goes on top, the NEW base

b

on the bottom. Writing it upside-down as

\dfrac{\log b}{\log a}

gives

\log_a b

, the reciprocal, and flips the whole answer.

The minimum is of the LOG, not the quadratic

After finding the quadratic's minimum (say

100

), you must still take

\log_{10} 100 = 2

. The smallest argument and the smallest log value are different numbers — the question asks for the log.

AP holding does not make it GP — test GP separately

p, q, r

being in AP says nothing about GP. The GP condition

q^2 = pr

must be checked on its own; for

\ln x, 3\ln x, 5\ln x

it fails because

9 \neq 5

, so the terms are AP-but-never-GP.

Solving Logarithmic Equations & Applications

Learn this subtopic in the notes

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

Common traps

$\log_{10} 0.2 = \log_{10} 2 - 1$ , which is negative

Writing

0.2 = \tfrac{2}{10}

gives

\log_{10} 0.2 = \log_{10} 2 - \log_{10} 10 = 0.3010 - 1 = -0.6990

. Dropping the

-1

flips the sign of

x

— the answer to

(0.2)^x = 2

is negative.

Throw out the non-positive t

A quadratic in

t = 2^x

often hands you a negative root. Because

2^x

is strictly positive, that root yields no real

x

— keep only

t > 0

before solving

x = \log_2 t

An algebraic root is not a solution until the domain clears it

x-1 = (x-3)^2

gives

x = 2

and

x = 5

, but

x = 2

makes

x-3 = -1 < 0

, so

\log_2(x-3)

is undefined. Only

x = 5

is a genuine solution — always re-substitute into the ORIGINAL equation.

The argument must be strictly positive — $0$ is not allowed

\log_a x

is undefined for

x \le 0

, and that includes

x = 0

: there is no power of

a

that gives

0

. A root that makes ANY argument equal to

0

(not just negative) must be rejected — the domain requires every argument

> 0

, strictly.

The GP condition squares the MIDDLE term

For

p, q, r

in GP the relation is

q^2 = pr

— the middle term is squared and set equal to the product of the outer two. Squaring the wrong term derails the whole chain-rule collapse.

Count the 5s, not the 2s — and don't forget 25, 125…

Each multiple of

25

contributes an EXTRA

5

beyond the one already counted by

\lfloor n/5\rfloor

. Stopping at

\lfloor n/5\rfloor

undercounts for

n \ge 25

Logarithms formulas

Logarithm Identities, Change of Base & Sums

What a Logarithm Is — Laws, Special Values, Domain

Applying the Laws — Evaluate and Combine

Change of Base & the Reciprocal Identity

Sign of a Logarithm & Bounds of a Log Function

Logarithms in AP/GP and the Geometric Mean

Common traps

Solving Logarithmic Equations & Applications

Taking the Log of an Exponential Equation

Substitution t = aˣ to a Quadratic

Domain Checks & Counting Solutions

GP, Chain-Rule & AM-GM Conditions

Application — Trailing Zeros of a Factorial

Common traps

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