Job 1• Pays £10 per day.

• Total Earnings• £300

• Pays 1p per day but daily pay increases by 50% each day.

• Total Earnings• £3835.00

Summer Jobs - 30 days work

Job 2

Exponential Functions

Aims: To know the general formula for an exponential graph.

What You Need To Be Able To Do

• To know the general formula for an exponential graph: We have seen the functions for linear, quadratic and some general polynomial functions.

• After this we will take a first look at logarithms.

Maths

• Ex 7A is very weird but maybe worth a look especially Q5

Geometric Sequence Graph

• The function of a basic exponential function f(x)=bx

• There are some limitations however…

Limits of b Values

• b must be positive (since –b would not be defined for fractional values of x with even denominators)

• b cannot be 1 since that would result in every value being 1 and therefore a line.

• So given this information lets see some exponential graphs…

• On a graphical calculator with a standard viewing window plot…

• y=2x y=4x y= (½)x

Look

–4 –2 2 4 6

2

4

6

x

y

Equation 2: y=4Ì

Equation 1: y=2Ì

Equation 3: y=(½)Ì

Look

–4 –2 2 4 6

2

4

6

x

y

Equation 2: y=0.5(4Ì)

Equation 1: y=3(2Ì)

Equation 3: y=4(½)Ì

Summary

• You are expected to be familiar with the shapes of exponential graphs.

• You are expected to know the effect of having different values of b for the graph y=bx.

• To know where these graphs cross the y axis and that this is the value of k in y=kbx.

• Lets have a quick go…

Sketch Showing Points of Intersection with the Axes.

xy 3

Sketch Showing Points of Intersection with the Axes.

xy 65

Sketch Showing Points of Intersection with the Axes.

xy 21

Sketch Showing Points of Intersection with the Axes.

xy 315

Sketch Showing Points of Intersection with the Axes.

xy 375

Sketch Showing Axes Intercepts and Asymptote

32 xy

Sketch Showing Points of Intersection with the Axes.

5102 xy

Sketch Showing Axes Intercepts

12 xy

Sketch Showing Points of Intersection with the Axes.

1b0 and 0agiven xbay

Logarithms

Aims: To know what logarithms are.

To be able to evaluate logarithms including solving equations

involving logarithms.

What You Need To Be Able To Do

• Name: What is a logarithm• Describe: The relationship between

an = x and Logax• Explain: How to solve basic missing

value type equations that include logs.

Maths

• Ex 7B p 356

Inversing

• What is the inverse of +?• What is the inverse of x?• What is the inverse of √?• But what is the inverse of taking 2 to

the power of a number e.g. How can you make x the subject of y=2x?

• The answer is that we do not currently have an inverse for an exponential…

Napier• John Napier of Merchiston (1550

– 4 April 1617) – also signed as Neper, Nepair – named Marvellous Merchiston, was a Scottish mathematician, physicist, astronomer & astrologer, and also the 8th Laird of Merchistoun.

• John Napier is most renowned as the inventor of the logarithm, and of an invention called "Napier's bones".

• Napier also made common the use of the decimal point in arithmetic and mathematics.

The Logarithm

• The logarithm is an inverse of an exponential base it is a function that must have a base corresponding to the base it is inversing.

• E.g. To inverse 2x one would have to use log base 2, written log2.

• We say that we apply logn to a value (a) and the answer is the power of n that gives you a.

• E.g. log216 = 4

The Logarithm Equivalence

• So IF 53=125 then Log5125=3

• Can you write this statement in general?

Solving Problems

• You will be expected to be able to write logarithm statements as indices and vice versa.

• E.g. If log61296 = 464=1296

Evaluating

• You need to be able to find the missing values in equations involving logarithms.

• Log4x = 3 what is x?

• Log2 1/16

= y what is y?

• Logx18 = 4 what is x?

• Log√xx3=a what is a?