Exponentials and Logarithms

Exponential Functions and Logarithms

Year 11 Methods

Introduction

• Exponential growth and decay are found in many practical situations, such as, finance, medicine, chemistry, earthquakes, computing, astronomy, sound, sports and marketing.

• Mathematics defines exponential functions as  $f\left(x\right)=ka^x$  

• where  $k$  is a non-zero constant and  $a$  is a positive real number other than 1

Exponents

Notice that index, power and exponent are interchangeable and that there is a powerful value difference between the operations:

2+4 = 6

2x4=8

2^4=16

Understanding Exponents

For more complex problems, you will need to expand the index form to find similar bases.

Multiple Choice

$$12a^3b^2$$  in expanded form is:

• $$12\times a\times a\times a\times b\times b$$  

• $$12\times3\times2\times a\times b$$  

Multiple Choice

What is the prime factorisation of 24?

• $$2\times12$$

• $$2\times2\times3$$

• $$2^3\times3$$

Index Laws

The index laws help simplify expressions and equations.

When using the laws, it is important that you:

- only use them when the bases are the same and

- that you do not multiply/divide, etc the bases.

Multiple Choice

Simplify the following: a⁰ + a⁰

• a⁰

• 2a⁰

• 1

• 2

Multiple Choice

a³ x a⁵

• a¹⁵
• a²
• a⁸
• a^-2

Multiple Choice

c⁵ ÷ c²

• c⁷
• c³
• c¹⁰
• c^-2

Multiple Choice

(y²)³

• y⁵
• y
• y²³
• y⁶

Multiple Choice

Simplify: $$\frac{a^5b^{10}}{a^3b^3}$$  

• $$a^8b^{13}$$  

• $$a^2b^7$$  

• $$a^{15}b^{30}$$  

Multiple Choice

Simplify: (4pq³)² ÷ (2p²q)³

• 2p^-4q³
• 8p⁴q³
• 16p⁴q²
• 4p^-4q²

Multiple Choice

Simplify: $$\frac{b^{5x}\times b^{2x+1}}{b^{3x}}$$  

• $$b^{4x+1}$$  

• $$b^{5x}$$  

Negative Indices

• Using the second index law we can see that a negative indices are equal to the reciprocal with a positive index.

• Always state the final answer as a positive index, unless otherwise stated.

Negative Index Law

Watch this video for conceptual understanding of the negative index law.

Multiple Choice

Simplify:
$$\left(3ab^2\right)\left(-2b^{-3}\right)$$  

• $$-\frac{6a}{b}$$  

• $$-\frac{6a^2}{b}$$  

• $$ab$$  

• $$\frac{3a}{2b}$$  

Multiple Choice

Which of the following is equivalent to the expression $$\frac{1}{2x^7}$$  ?

• $$\left(2x\right)^{-7}$$  

• $$\frac{1}{2}x^7$$  

• $$\frac{1}{2}x^{-7}$$  

• $$2x^{-7}$$  

Multiple Choice

(y⁹)^-2

• y¹⁸
• y¹¹
• 1/y¹⁸
• y⁹/2

Multiple Choice

How else could you write $$4^{-2}$$  ?

• $$16$$  

• $$-16$$  

• $$-\frac{1}{16}$$  

• $$\frac{1}{16}$$  

Composite Bases

Composite bases can be factorised into primes to simplify the calculations.

Check out the video on prime factorisation to recap!

Simplifying using prime factorisation

Consider the folllowing expressions:

 $\frac{12^3}{4^3}$  (Using prime factorisation and the index laws we could simplify)

 $\frac{\left(3\times4\right)^3}{4^3}$  

 $\frac{3^3\times4^3}{4^3}$  

 $3^3=27$