Additional Mathematics notes, chapter 4

Indices, Surds and Logarithms

Textbook Additional Mathematics Form 4 KSSM (DLP), pages 88–125

What this chapter covers

Simplify expressions with indices and solve equations involving indices using the laws of indices, including negative and rational indices (4.1)
Distinguish rational and irrational numbers, use the laws of surds, rationalise denominators and solve problems (4.2)
Relate logarithms to indices, use the laws of logarithms and change of base, and solve equations with logarithms (4.3)
Solve real-life problems of growth and decay using indices, surds and logarithms (4.4)

4.1Laws of indices

In aⁿ, a is the base and n is the index (power). The laws below hold for any non-zero base and are used to simplify algebraic expressions too.

Law	Example
a^m × aⁿ = a^{m + n}	t² × t³ = t⁵
a^m ÷ aⁿ = a^{m − n}	x⁷ ÷ x⁴ = x³
(a^m)ⁿ = a^mn	(y³)² = y⁶
a⁰ = 1	7⁰ = 1
a⁻ⁿ = 1aⁿ	2⁻³ = 18
a¹ⁿ = ⁿ√a	8¹³ = ³√8 = 2
a^mn = (ⁿ√a)^m = ⁿ√(a^m)	8²³ = 2² = 4

Simplify (x³y⁻²)²x y⁻¹

Numerator: x⁶y⁻⁴. Divide: x⁶⁻¹y^{−4−(−1)} = x⁵y⁻³ = x⁵y³.

For a rational index, take the root first, then the power: 27²³ = (³√27)² = 3² = 9. The numbers stay small.

(a + b)ⁿ is not aⁿ + bⁿ, and 2³ × 3³ cannot be combined by adding indices because the bases differ.

Equations involving indices

Write both sides with the same base, then equate the indices: if a^x = a^y then x = y.

Solve 4^x+1 = 32

2^2(x+1) = 2⁵, so 2x + 2 = 5 and x = 32.
When a sum appears, factorise: 3^x+1 + 3^x = 36 gives 3^x(3 + 1) = 36, 3^x = 9, x = 2.

Simplify (a) 16⁻³⁴ (b) solve 9^x = 27^x−1

Show answerHide answer

(a) 16^−3/4 = 1 ÷ 2³ = 18
(b) 3^2x = 3^3x−3, so 2x = 3x − 3 and x = 3

4.2Laws of surds

A rational number can be written as ab with a, b integers, b ≠ 0 (its decimal ends or recurs). An irrational number cannot. A surd is an irrational root such as √2 or ³√5.

√9 = 3 is rational, so it is not a surd. √7 and √12 are surds.

Law of surds	Example
√a × √b = √(ab)	√2 × √8 = √16 = 4
√a ÷ √b = √(ab)	√30 ÷ √5 = √6
(√a)² = √(a²) = a	(√7)² = 7
m√a + n√a = (m + n)√a	3√5 + 2√5 = 5√5
√(a²b) = a√b	√48 = √(16 × 3) = 4√3

√a + √b is not √(a + b). For example √9 + √16 = 3 + 4 = 7, but √(9 + 16) = √25 = 5.

Rationalising the denominator

Remove the surd from the denominator by multiplying top and bottom by the right factor.

Denominator	Multiply by	Example
√a	√a√a	6√3 = 6√33 = 2√3
a + √b	conjugate a − √ba − √b	13 + √5 = 3 − √59 − 5 = 3 − √54
√a − √b	conjugate √a + √b√a + √b	2√5 − √3 = 2(√5 + √3)5 − 3 = √5 + √3

The conjugate of a + √b is a − √b. Their product (a + √b)(a − √b) = a² − b is always rational.

Express 2 + √32 − √3 in the form p + q√3

(2 + √3)(2 + √3)(2 − √3)(2 + √3) = 4 + 4√3 + 34 − 3 = 7 + 4√3.

Simplify (a) √50 − √18 (b) rationalise 4√2 + 1

Show answerHide answer

(a) 5√2 − 3√2 = 2√2
(b) 4(√2 − 1)2 − 1 = 4√2 − 4

4.3Laws of logarithms

Logarithm. a^x = N is the same statement as log_a N = x, with a > 0, a ≠ 1 and N > 0. A logarithm is an index.

Index form	Logarithm form
2⁵ = 32	log₂ 32 = 5
10⁻² = 0.01	log₁₀ 0.01 = −2

Special values: log_a 1 = 0, log_a a = 1. log₁₀ is written lg (or log), and log_e is the natural logarithm ln, where e ≈ 2.718. Note ln e = 1 and e^{ln x} = x.

Graphs of y = 2 to the power x and y = log base 2 of x, mirror images in the line y = x; points (0,1) and (1,0), (1,2) and (2,1) are marked — y = 2^x and y = log₂ x are reflections of each other in y = x, because one undoes the other.

The three basic laws (a, x, y > 0 and a ≠ 1)

Law	Rule
Product	log_a xy = log_a x + log_a y
Division	log_a xy = log_a x − log_a y
Power	log_a xⁿ = n log_a x

log_a(x + y) cannot be split, and log_a xlog_a y is not log_a x − log_a y. Also, all the logs in a law must have the same base.

Given log₂ 3 = 1.585 and log₂ 5 = 2.322, find log₂ 15 and log₂ 0.6

log₂ 15 = log₂ 3 + log₂ 5 = 3.907.
log₂ 0.6 = log₂ 35 = 1.585 − 2.322 = −0.737.

Change of base

log_a b = log_c blog_c a. Special case: log_a b = 1log_b a.

Find log₄ 8 without a calculator, and log₃₀ 4 with one

log₄ 8 = log₂ 8log₂ 4 = 32.
log₃₀ 4 = lg 4lg 30 = 0.60211.4771 = 0.408.

Equations involving indices and logarithms

Solve 5^x = 12 and log₃(x − 1) = 2

Take lg of both sides: x lg 5 = lg 12, so x = 1.07920.6990 = 1.544.
log₃(x − 1) = 2 means x − 1 = 3² = 9, so x = 10.
Equations with several logs: combine them with the laws, then convert to index form, and reject any answer that makes a log argument zero or negative.

Given log_a 2 = p and log_a 3 = q, express log_a 18 in terms of p and q

Show answerHide answer

log_a 18 = log_a(2 × 3²) = p + 2q

4.4Applications

Growth, decay and compound interest use quantities such as P(1 + r)ⁿ. To find an unknown index n, take logarithms.

When will RM2 000 at 7% a year exceed RM4 000?

2 000(1.07)ⁿ > 4 000, so 1.07ⁿ > 2 and n > lg 2lg 1.07 = 0.301030.02938 = 10.24.
Years are whole numbers, so the minimum is 11 years.

When you divide both sides by a log, check its sign: if it is negative the inequality sign reverses. For growth, lg 1.07 > 0, so it stays.

Mr Chong has RM4 000(1.1)ⁿ after n years. Find the first whole n so that it exceeds RM5 100.

Show answerHide answer

1.1ⁿ > 1.275, n > lg 1.275 ÷ lg 1.1 = 0.10551 ÷ 0.04139 = 2.55, so n = 3 years