Logarithm

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What: Logarithm covers the definition $\log_b a = c \iff b^c = a$, the six fundamental logarithm laws, change of base, and CDS's typical simplification, equation, and characteristic/mantissa problems.
Why it matters: CDS papers from 2009 to 2023 average 2–3 questions per sitting — small chapter, high return per minute, formulaic answers.
Key fact: $\log_a a = 1$, $\log_a 1 = 0$, $\log a^n = n \log a$, $\log ab = \log a + \log b$, $\log (a/b) = \log a - \log b$, and the change-of-base $\log_b a = \log a / \log b$. Six identities cover 90% of CDS logarithm questions.

Logarithm is the inverse of exponentiation — and one of the most "formulaic" chapters in CDS arithmetic. Once you have the six laws on tap, almost every question is one-line algebra. The high-value moves are converting products to sums (and quotients to differences) before you compute.

This page is built from CDS Previous Year Questions across 2009–2023. Pair with Power and Roots (the inverse direction) and SI/CI (logarithm gives the doubling time for CI).

What This Topic Covers

CDS scope: (1) definition — $\log_b a = c \iff b^c = a$; (2) the six laws; (3) change of base; (4) common logarithms (base 10) and natural logarithms (base $e$) — CDS uses base 10 almost exclusively; (5) characteristic and mantissa; and (6) simplification of logarithmic expressions.

Why This Topic Matters

2–3 CDS questions per paper, all formula-driven.
Connects directly to Power and Roots (inverse operations).
Used in compound-interest doubling-time and exponential-growth reasoning.

Exam Pattern & Weightage

Year / Paper	No.	Subtopics Tested
2009-II	2	Basic law application, simplification
2011-I/II	2	Change of base, log equation
2012-I/II	2	Product to sum, basic identity
2016-I/II	3	Simplification, log values
2017-I/II	2	Equation in $x$, product of logs
2019-II	2	Change of base, value finding
2020-I/II	2	Log laws, simplification
2021-I/II	2	Equation, value
2022-I / 2023-I	2	Mixed simplification

⚡ CDS Alert

The change-of-base trick $\log_b a = \log_k a / \log_k b$ for any positive base $k \neq 1$ is the single most-tested logarithm identity in CDS. Choose $k = 10$ when log tables (or "given $\log 2 = 0.301$") are available; $k = a$ or $k = b$ often gives a clean form.

Core Concepts

Definition

Logarithm Definition $$\log_b a = c \iff b^c = a \quad (a > 0,\, b > 0,\, b \neq 1)$$

The Six Fundamental Laws

Laws of Logarithms $$\log_b 1 = 0 \qquad \log_b b = 1 \qquad b^{\log_b a} = a$$ $$\log_b(xy) = \log_b x + \log_b y \qquad \log_b\!\left(\frac{x}{y}\right) = \log_b x - \log_b y$$ $$\log_b(x^n) = n \log_b x$$

Change of Base

Change of Base $$\log_b a = \frac{\log_k a}{\log_k b} \quad \text{for any positive } k \neq 1$$

Particular cases: $\log_b a = 1/\log_a b$ (reciprocal); $\log_b a = \log_b c \cdot \log_c a$ (chain).

Common Logarithm Values to Memorise

Standard log values (base 10) $\log 2 \approx 0.301,\quad \log 3 \approx 0.477,\quad \log 5 \approx 0.699,\quad \log 7 \approx 0.845$

From these, derive: $\log 4 = 2 \log 2 = 0.602$, $\log 6 = \log 2 + \log 3 = 0.778$, $\log 8 = 3 \log 2 = 0.903$, $\log 9 = 2 \log 3 = 0.954$, $\log 10 = 1$.

Characteristic and Mantissa

For a common (base-10) logarithm of a positive number $N$, the characteristic is the integer part of $\log N$, and the mantissa is the (always positive) fractional part. If $N$ has $n$ digits, the characteristic is $n - 1$. Example: $\log 345 \approx 2.5378$; characteristic = 2 (so 345 has 3 digits).

Logarithmic Equations

To solve $\log_b x = c$: rewrite as $x = b^c$. To solve $\log_b f(x) = \log_b g(x)$: set $f(x) = g(x)$ (and check domain).

Worked Examples

Example 1 — Direct Application (2009-II)

Q: If $\log_{10} 2 = 0.301$, find $\log_{10} 16$.

$\log 16 = \log 2^4 = 4 \log 2 = 4 \cdot 0.301 = 1.204$.

Example 2 — Product to Sum (2012-I)

Q: Simplify $\log 12 - \log 6 + \log 5 - \log 10$.

Combine: $\log(12/6) + \log(5/10) = \log 2 + \log(1/2) = \log 1 = 0$.

Example 3 — Change of Base (2019-II)

Q: If $\log_4 x = 2$, find $x$.

By definition: $x = 4^2 = 16$.

Example 4 — Logarithm Equation (2017-II)

Q: Solve $\log_3(x + 2) = 2$.

Rewrite: $x + 2 = 3^2 = 9 \implies x = 7$.

Example 5 — Sum of Logs (2020-I)

Q: Simplify $\log 25 + \log 4$.

$\log 25 + \log 4 = \log(25 \cdot 4) = \log 100 = 2$.

Example 6 — Reciprocal Identity (2021-I)

Q: Find $\log_3 27$ if $\log_{27} 3 = 1/3$.

Use reciprocal: $\log_3 27 = 1/\log_{27} 3 = 1/(1/3) = 3$.
Verify: $3^3 = 27$. ✓

Example 7 — Characteristic (2016-I)

Q: The number of digits in $2^{50}$ is given to satisfy $\log_{10} 2 = 0.301$. Find it.

$\log 2^{50} = 50 \cdot 0.301 = 15.05$. Characteristic = 15, so $2^{50}$ has 16 digits.

How CDS Tests This Topic

Five archetypes: (1) compute a logarithm given standard values, (2) simplify a sum/difference of logarithms, (3) solve a logarithmic equation, (4) apply change of base or reciprocal identity, (5) find the number of digits using characteristic. Each takes 30–60 seconds.

Exam Shortcuts (Pro-Tips)

Shortcut 1 — Memorise the Standard Values

$\log 2 = 0.301$, $\log 3 = 0.477$, $\log 5 = 0.699$, $\log 7 = 0.845$, $\log 10 = 1$. Build everything else from these.

Shortcut 2 — Combine Before Computing

For an expression like $\log a + \log b - \log c$, always combine first to $\log(ab/c)$, then evaluate.

Shortcut 3 — Reciprocal Identity

$\log_a b = 1/\log_b a$. Useful when one direction is "ugly" and the other is clean.

Shortcut 4 — Number of Digits

The number of digits in $N$ (where $N$ is a positive integer) is $\lfloor \log_{10} N \rfloor + 1$.

Shortcut 5 — Power and Log Together

$b^{\log_b a} = a$ — the cleanest identity. Often hidden inside CDS problems disguised as something more complex.

Common Question Patterns

Pattern 1 — Direct Computation

Given standard logs, find $\log N$ for some $N$ via the six laws.

Pattern 2 — Simplification

Combine multiple logs into one. Apply product/quotient laws.

Pattern 3 — Equation

$\log_b x = c$ ⇒ $x = b^c$. Always check the domain ($x > 0$).

Pattern 4 — Change of Base / Reciprocal

Convert to a base where the answer is clean, or use $\log_a b = 1/\log_b a$.

Pattern 5 — Number of Digits

Compute $\log_{10} N$; take the floor; add 1.

Preparation Strategy

Week 1. Memorise the six laws and the standard log values ($\log 2, 3, 5, 7$). Drill 20 problems on direct application and simplification. Master the "combine first, then evaluate" reflex.

Week 2. Change of base, reciprocal identity, and characteristic. Practice number-of-digits problems. Cross-train with Power and Roots — every log identity has an exponent identity counterpart.

Mock testing. Use CDS mock tests. Logarithm is short and clean — once the laws are reflex, the questions resolve themselves.

Drill Logarithm at Speed

CDS mocks with simplification, change of base, and number-of-digits problems. Six laws — five archetypes — reflex.

Start Free Mock Test

Frequently Asked Questions

What does $\log_b a$ mean?

It is the exponent to which the base $b$ must be raised to give $a$. So $\log_b a = c \iff b^c = a$. Logarithm and exponentiation are inverse operations.

What are the six fundamental logarithm laws?

$\log_b 1 = 0$, $\log_b b = 1$, $b^{\log_b a} = a$, $\log_b(xy) = \log_b x + \log_b y$, $\log_b(x/y) = \log_b x - \log_b y$, $\log_b(x^n) = n \log_b x$. These six cover 90% of CDS logarithm questions.

What is the change-of-base formula?

$\log_b a = \log_k a / \log_k b$ for any positive base $k \neq 1$. Particular cases: $\log_b a = 1/\log_a b$ (reciprocal). Use whenever the natural base of a problem isn't the one you have values for.

How do I find the number of digits in a large number using logs?

The number of digits is $\lfloor \log_{10} N \rfloor + 1$. Example: digits in $2^{50}$. $\log 2^{50} = 50 \cdot 0.301 = 15.05$. Floor = 15. Number of digits = 16.

What are characteristic and mantissa?

For $\log_{10} N$ (positive $N$), the characteristic is the integer part and the mantissa is the (positive) fractional part. $\log 345 \approx 2.5378$ has characteristic 2 (so $345$ has 3 digits) and mantissa 0.5378.

What standard log values should I memorise?

$\log 2 \approx 0.301$, $\log 3 \approx 0.477$, $\log 5 = 1 - \log 2 \approx 0.699$, $\log 7 \approx 0.845$. From these, build: $\log 4 = 0.602$, $\log 6 = 0.778$, $\log 8 = 0.903$, $\log 9 = 0.954$, $\log 10 = 1$, etc.

Which CDS Maths topics connect to Logarithm?

Power and Roots — the inverse operation. SI/CI — logarithm gives the doubling time for compound interest. Quadratic equations occasionally involve logs in transformations.