CDS Mathematics · Logarithms

Laws of Logarithms

A logarithm is an exponent: log turns products into sums, powers into multiples, and any number into a combination of log 2, log 3 and log 5.

All Logarithms notes CDS Mathematics strategy

Why this matters

Thirteen PYQs, two of them HARD. Almost every one is solved by breaking each number into primes and powers of 10 — 31.25 = 10³/2⁵, 384 = 2⁷·3 — then adding logs. Three facts finish the rest: log 1 = 0, log 10 = 1, and log 2 + log 5 = 1.

Concept 1 of 1: Product, power and change of base

Since

\log_{10}N

is the power of

10

that gives

N

, multiplying numbers adds their powers and raising to a power multiplies it. So every log reduces to logs of primes plus whole numbers from the powers of

10

Definition

$\log(ab) = \log a + \log b$ ; $\log\dfrac ab = \log a - \log b$ ; $\log a^n = n\log a$ .
$\log_{10}5 = 1 - \log_{10}2$ ; a factor of $10^k$ adds $k$ .
Change of base: $\log_b a = \dfrac{\log a}{\log b}$ ; so $\log_{100}x = \tfrac12\log_{10}x$ .
A negative log written with a bar: $-0.0714 = \overline{1}.9286$ (characteristic $-1$ , positive mantissa).
For $0 < m < 1$ : $\log m < 0$ .

Laws

\log(ab) = \log a + \log b, \qquad \log a^n = n\log a

Worked example

Write

\log_{10}62.5

in terms of

\log_{10}2

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2021 · CDS (I) 2021 — Elementary Mathematics · Q41Moderate

Example 1 · Logarithms · Logarithm Identities and Change of Base

What is

\log_{10} 31.25

equal to?

The bar applies to the characteristic only

\overline{1}.9286

means

-1 + 0.9286 = -0.0714

, not

-1.9286

. Writing negative logs this way keeps the mantissa positive.

Drill 12 more on product, power and change of base

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Product, power and change of base
Laws
$\log(ab) = \log a + \log b, \qquad \log a^n = n\log a$

Watch out for (1)

The bar applies to the characteristic only
→ Product, power and change of base

Test yourself on Logarithms

15 past CDS questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Logarithms chapter test Drill all 13 questions