Logarithms; Simplification & BODMAS; Partnership

Logarithms

26.1 Definition

logₐ(b) = c means aᶜ = b

Logarithm answers: "To what power must the base be raised to get b?"

26.2 Key Properties

Property	Formula
Product	log(mn) = log(m) + log(n)
Quotient	log(m/n) = log(m) - log(n)
Power	log(mⁿ) = n × log(m)
Base change	logₐ(b) = log(b)/log(a)
Identity	logₐ(a) = 1
Zero	logₐ(1) = 0
Reciprocal base	logₐ(b) = 1/log_b(a)

26.3 Common Logarithm vs Natural Logarithm

Common log: log₁₀ (base 10), written as log
Natural log: logₑ (base e), written as ln, where e ≈ 2.71828

Simplification & BODMAS

27.1 BODMAS Rule

Operations must be performed in this order:

Letter	Operation
B	Brackets — solve innermost first
O	Of / Orders (powers, roots)
D	Division
M	Multiplication
A	Addition
S	Subtraction

27.2 Types of Brackets (in order)

( ) — Simple/Round brackets — solve first
{ } — Curly brackets — solve second
[ ] — Square brackets — solve last

27.3 Shortcuts in Simplification

Fraction comparisons:

To compare a/b and c/d: Compare ad vs bc
If ad > bc → a/b > c/d

Simplifying large expressions:

Use algebraic identities to factor
Look for patterns like (a² - b²) = (a+b)(a-b)

Approximations:

√2 ≈ 1.414
√3 ≈ 1.732
√5 ≈ 2.236
π ≈ 3.14159
(1 + 1/n)ⁿ ≈ e for large n

Partnership

28.1 Concept

When two or more persons invest in a business:

Profit/Loss shared in ratio of their investments × time

28.2 Simple Partnership (Same Time Period)

If A invests 'a' and B invests 'b' for the same duration:

A's profit : B's profit = a : b

28.3 Compound Partnership (Different Time Periods)

If A invests 'a' for 't₁' time and B invests 'b' for 't₂' time:

A's profit : B's profit = (a × t₁) : (b × t₂)

28.4 Working Partner vs. Sleeping Partner

Working partner may get a salary first, remainder split by investment ratio

Sleeping partner only shares profit by investment ratio

28.5 Example

A invests Rs. 5000 for 12 months, B invests Rs. 4000 for 8 months. Total profit = Rs. 2800:

A's ratio = 5000 × 12 = 60,000
B's ratio = 4000 × 8 = 32,000
Ratio = 60:32 = 15:8

A's profit = 2800 × 15/23 = Rs. 1826 (approx)

B's profit = 2800 × 8/23 = Rs. 974 (approx)

📊 Master Formula Sheet

Interest

SI = PRT/100
CI = P(1 + R/100)ⁿ - P
CI - SI (2 years) = P(R/100)²
CI - SI (3 years) = P(R/100)²(3 + R/100)

Work and Time

Together = ab/(a+b)
MDH/W = constant

Speed, Distance, Time

D = S × T
km/hr to m/s = × 5/18
Average speed (equal dist) = 2S₁S₂/(S₁+S₂)

Profit and Loss

Profit% = (Profit/CP) × 100
SP = CP(100+P%)/100
CP = SP × 100/(100+P%)
False weight profit% = Error/(True-Error) × 100

Percentage

Net change (two successive) = x + y + xy/100

Permutation and Combination

nPr = n!/(n-r)!
nCr = n!/[r!(n-r)!]
Circular = (n-1)!

Geometry

Area of triangle = √[s(s-a)(s-b)(s-c)]
Equilateral triangle = (√3/4)×a²
Circle area = πr²
Sphere volume = (4/3)πr³
Cone volume = (1/3)πr²h