Topic 6: Indices, Exponentials, Logarithms and Surds (Chapter 2)

Index Law 1

a¹ = a.

Index Law 2

a^m × a^n = a^(m+n).

Index Law 3

a^m ÷ a^n = a^(m-n).

Index Law 4

(a^m)^n = a^(m×n).

Index Law 5

(ab)^m = a^m × b^m.

Index Law 6

(a/b)^m = a^m / b^m.

Zero Index

a⁰ = 1 (a ≠ 0).

Negative Indices

a^(-m) = 1 / a^m.

Fractional Indices

a^(1/n) = n√a; a^(m/n) = (n√a)^m.

Rational Numbers

Expressed as fraction of two integers.

Irrational Numbers

Cannot be expressed as simple fraction; decimals non-repeating/non-terminating.

Surds

Irrational roots, e.g., √2.

Like Surds

Multiples of same surd; can be added/subtracted.

Simplifying Surds

√x × √y = √(xy).

Rationalising Denominator

x / √y = (x√y) / y.

Binomial Products with Surds

Expand using distributive law (FOIL).

Definition of a logarithm

log_b(a) = c means b^c = a

Common logarithm

Base 10 logarithm: log = log₁₀

Natural logarithm

ln = log base e

Change of base formula

logb(a) = logc(a) / log_c(b)

log_b(b) = ?

log_b(b) = 1

log_b(1) = ?

log_b(1) = 0

log_b(a^n) = ?

n × log_b(a)

logb(a) + logb(c) = ?

log_b(a × c)

logb(a) - logb(c) = ?

log_b(a ÷ c)

log_b(1/a) = ?

-log_b(a)

log_b(√a) = ?

(½) × log_b(a)

Exponential form to log

3^x = 81 → log₃(81) = x

Solve: log₅(x) = 3

x = 5^3 = 125

If log_b(x) = y →

x = b^y

Inverse of exponential

Logarithmic function

Domain of log(x)

x > 0

Range of log(x)

All real numbers

Graph of y = log(x)

Passes through (1,0); vertical asymptote at x = 0

Index Law 1 (Multiplying)

a^m × a^n = a^(m+n)

Index Law 2 (Dividing)

a^m ÷ a^n = a^(m−n)

Index Law 3 (Power of a power)

(a^m)^n = a^(mn)

Zero index

a^0 = 1 (a ≠ 0)

Negative index

a^(−n) = 1 / a^n

Fractional index

a^(m/n) = ⁿ√(a^m) = (ⁿ√a)^m

Surd

An irrational root (e.g., √2, ³√5)

Simplifying surds

√ab = √a × √b

Rationalising denominator (simple)

1 / √a = √a / a

Rationalising denominator (binomial)

Multiply by conjugate: (a + √b)(a − √b)

Exponential equation

y = a^x (growth/decay depending on a)

Exponential growth

y = a^x, where a > 1

Exponential decay

y = a^x, where 0 < a < 1

Logarithm definition

logₐx = y ↔ a^y = x

logₐ(a)

1

logₐ(1)

0

logₐ(a^x)

x

a^(logₐx)

x

Change of base rule

logₐb = logc(b) / logc(a)

Solving log equations

Rewrite in exponential form or use log rules

Log rule 1 (Product)

logₐ(xy) = logₐx + logₐy

Log rule 2 (Quotient)

logₐ(x/y) = logₐx − logₐy

Log rule 3 (Power)

logₐ(x^n) = n × logₐx