Sine graph

Cosine graph

Tangent graph

Now, try these questions:

Some examples of graph of logarithms, taken from purplemaths:

y = ln(x)	y = ln(–x)	y = –ln(x)
y = ln(x + 1)	y = ln(–x + 1)	y = –ln(–x)
y = ln(x) – 1	y = ln(–x) – 1	y = ln(x – 1)
y = ln(x) + 1	y = ln(–x) + 1	y = ln(–x – 1)

What is a natural logarithms?

Logarithms derived to base e are called natural logs and are often denoted “ln” rather than “log”.

The following video clip is about

1. properties of natural logarithms
2. solving a natural logarithmic equation
3. solving an exponential equation

Application of natural logarithms

Natural logarithms are commonly used throughout science and engineering. For example, see:  Application of Natural Logarithm.

the meaning of e

Practice Questions

Laws of Logarithms

Logs have some very useful properties which follow from their definition and the equivalence of the logarithmic form and exponential form.  Some useful properties are as follows:

Answer

Other Laws of Logarithms

Click on resource attached, to see examples and practice questions.

Extension

Harder Logarithm Questions

Another law ….

Example:

Worked examples:

With the help of the video below, complete the Skill Practice questions:

To solve exponential equations, you need to have equations with comparable exponential expressions on either side of the “equals” sign, so you can compare the powers and solve.

This video clip below demonstrated exponential equations are solved. Make some notes in your workbook while watching the video clip.

Skill Practice

Scroll down the page until you see Examples.  Complete the questions at the bottom of the page.  Remember to check you answer.  (Note:  Ignore the last question.)

Solving Exponential Inequations

Skill Practice

Law of Rational Exponent:

A useful website for your reference, when completing the skill practice task below.

Skill Practice

1. Try these questions. Check your answer after attempting the questions.

2. Complete worksheet attached.

Index and base form

Let’s look at 100 = 10²

• the 2 is called the index number
• the 10 is called the base number

Another name for index form is power form or power notation.

Watch the video below and write down the general rules for index laws in your booklet.

Skill practice – Scroll down the page.  Complete “Have a Go” and “Practice Questions”.  Record your responses in your workbook.

Note:

An algebraic expression involving a number that is raised to the power of another number is known as an indicial or exponential expression.  

Prime Factor and Factor Trees

In order to simplify indicial expressions fully, we may need to express exponential terms with the same base.  In order to do this, we need to find the prime factors of numbers.  We can use factor trees to evaluate the prime factors of many numbers.

For example, the number 504 can be expressed as 2³ x 3² x 7:

A worked example:

Practice Questions

Complete the following questions.  Remember to show your working out in your workbook.

Twelve students are to have their photos taken. How many ways are there of arranging the group if they all sit in a row?

Solution:

As there are twelve students the number of arrangements is

12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 470 001 600

which is rather a large number of choices!

So for convenience this can be written as 12! which is read as “12 factorial”.

Generalisation:

Not all arrangements require the use of every object available, as is shown in the following video clip:

Generalisation:

Permutations

Combinations

Arrangements in a circle

Consider the case of seating five people at a round table.  In how many different ways can this be done?

If we let the 5 people be represented by a, b, c, d and e we can see that the linear arrrangements a b c d e and b c d e a are different, but in a circle they both become

In fact a b c d e,  b c d e a,  c d e a b,  d e a b c and e a b c d all result in the same circular arrangement.

Task:

Worksheet: Complete the questions and record your responses in your workbook. Answers are attached, try the questions first and check you responses with the answers given.

CAS calculator

Calculating factorials

Example: To calculate 10!

Enter 10 then find the factorial symbol in the Probability menu (press “menu” – “probability” – “factorial (!)”). The result then appears when you press “enter”.

Finding permutations and combinations

Example: To find 11P7 and 5C3.

1. The required syntax is nPr(n,r) or nCr(n,r). You will find both of these in the probability menu. Press “enter” – “probability” – “permutations”/”combinations”. To find the required answers fill in the values as required, eg.

nPr(11, 7) and nCr(5, 3)