Maths Methods CAS @ CSHS

Introduce Types of Logarithms

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

Formula to change the base:

Quick Review:

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:

Solving Exponential Equations

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

workshop: discovering-the-laws-of-rational-exponents

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.

Some numbers can be written in mathematical shorthand if the number is the product of ‘repeating numbers’.  For example, 100 = 10 x 10, can be written in shorthand as 10².

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.