**Indices and Logarithms**

**Indices**

Definition - Any expression written as an is defined as the variable a raised to the power of the number n

n is called a power, an index or an exponent of a

Example - where n is a positive whole number,

a1 = a

a2 = a a

a3 = a a a

an = a a a a……n times

Detail 1

Rule 1

Rule 1 am. an = am+n

e.g. 22 . 23 = 25 = 32

e.g. 51 . 51 = 52 = 25

e.g. 51 . 50 = 51 = 5

Detail 3

Detail 4

rewrite equation so x is not a power

Take logs of both sides

log(5x) = log(23x)

rule 1 => log 5x = log 2 + log 3x

rule 3 => x.log 5 = log 2 + x.log 3

Cont……..

Another Example:

Find the value of x

5x = 2(3)x

All logs must be to the same base in applying the rules and solving for values

The most common base for logarithms are logs to the base 10, or logs to the base e (e = 2.718281…)

Logs to the base e are called Natural Logarithms

logex = ln x

If y = exp(x) = ex

then loge y = x or ln y = x

A Note of Caution:

Definition - Any expression written as an is defined as the variable a raised to the power of the number n

n is called a power, an index or an exponent of a

Example - where n is a positive whole number,

a1 = a

a2 = a a

a3 = a a a

an = a a a a……n times

Indices

The following rules of logs apply

Logarithms

Jacques Text Book (edition 4):

Section 2.3 & 2.4

Indices & Logarithms

Topic 4: Indices and Logarithms

Up to students to revise and practice the rules of indices and logs using examples from textbooks.

These rules are very important for remaining topics in the course.

Good Learning Strategy!

non-linear

always positive

as x get

y and

slope of graph (gets steeper)

Features of y = ex

Evaluate the following:

Rule 1 am. an = am+n

e.g. 22 . 23 = 25 = 32

e.g. 51 . 51 = 52 = 25

e.g. 51 . 50 = 51 = 5

Rule 2

All indices satisfy the following rules in mathematical applications

2.

3.

4.

)

1

And……..

)

1

1

From the above rules, it follows that

These are practice questions for you to try at home!

Simplify the following using the above Rules:

Rule 2 notes…

3) A Zero power

a0 = 1

e.g. 80 = 1

4) A Fractional power

e.g.

An Example : Find the value of x

(4)x = 64

1) rewrite equation so that it is no longer a power

Take logs of both sides

log(4)x = log(64)

rule 3 => x.log(4) = log(64)

2) Solve for x

x =

Does not matter what base we evaluate the logs, providing the same base is applied both to the top and bottom of the equation

3) Find the value of x by evaluating logs using (for example) base 10

x = ~= 3

Check the solution

(4)3 = 64

Logs can be used to solve algebraic equations where the unknown variable appears as a power

1) where n is positive whole number

an = a a a a……n times

e.g. 23 = 2 2 2 = 8

2) Negative powers…..

a-n =

e.g. a-2 =

e.g. where a = 2

2-1 = or 2-2 =

Indices satisfy the following rules:

An Example : Find the value of x

200(1.1)x = 20000

Simplify

divide across by 200

(1.1)x = 100

to find x, rewrite equation so that it is no longer a power

Take logs of both sides

log(1.1)x = log(100)

rule 3 => x.log(1.1) = log(100)

Solve for x

x =

no matter what base we evaluate the logs, providing the same base is applied both to the top and bottom of the equation

Find the value of x by evaluating logs using (for example) base 10

x = = 48.32

Check the solution

200(1.1)x = 20000

200(1.1)48.32 = 20004

Logs can be used to solve algebraic equations where the unknown variable appears as a power

Jacques Text Book (edition 4):

Section 2.3 & 2.4

Indices & Logarithms

Topic 4: Indices and Logarithms

1) where n is positive whole number

an = a a a a……n times

e.g. 23 = 2 2 2 = 8

2) Negative powers…..

a-n =

e.g. a-2 =

e.g. where a = 2

2-1 = or 2-2 =

Indices satisfy the following rules: