Block 3

Tricky log Graphs

What is to be learned?

• How to draw and recognise nasty log graphs

Previously y = log4x

y = 4x y = x

y = log4x1

1

y = log4x

y = log4x

1

(4, 1)

( , 0) x y

y = log41

y = 0 y = log44y = 1

x = 1 x = 4

(1 , 0) (4 , 1)

Mini log rules are very helpful

Draw y = 4 log5xsub x = 1 and

x = 5x=1 y = 4log51

= 4 X 0 (1 , 0)

x=5 y =4 log55

= 0

(5 , 4)= 4 = 4 X 1

(5, 4)

(1, 0)

y = 4 log5x

Draw y = 6 log7xsub x = 1 and

x = 7x=1 y = 6log71

= 6 X 0 (1 , 0)

x=7 y =6 log77

= 0

(7 , 6)= 6 = 6 X 1

(7, 6)

(1, 0)

y = 6 log7x

Draw y = 2 log3(x – 1) sub x = 2 and x = 4

x=2 y = 2log3(2 – 1)= 2 X log31

(2 , 0)x=4 y = 2log3(4 – 1)

= 2 X 0

(4 , 2)= 2 X 1 = 2 X log33

(4, 2)

(2, 0)y = 2 log3 (x – 1)

want to equal 1 and 3

= 0

= 2

Sketching Nasty Log GraphsTactics

Find 2 points using mini log rulesi.e. chose x values to make

loga1 and

logaa

Draw y = 5log4(x – 2) sub x = 3 and x = 6

x = 3 y = 5log4(3 – 2)= 5 X log41

(3 , 0)x = 6 y = 5log4(6 – 2)

= 5 X 0

(6 , 5)= 5 X 1 = 5 X log44

(6, 5)

(3, 0)

y = 5 log4 (x – 2)

want to equal 1 and 4

= 0

= 5

Draw y = 6 log3(x + 2) sub x = -1 and x = 1

x=-1 y = 6log3(-1 + 2)= 6 X log31

(-1 , 0)x=1 y = 6log3(1 + 2)

= 6 X 0

(1 , 6)= 6 X 1 = 6 X log33

(1, 6)

(-1, 0)y = 6 log3 (x + 2)

want to equal 1 and 3

= 0

= 6

Key Question

Identifying Log Graphs

(6, 7)

(2, 0)

Type y = a log5 (x + b)

?

Identifying Log Graphs

(6, 7)

(2, 0)

Type y = a log5 (x + b)

x y

x = 2, y = 0→ 0 = a log5 (2 + b)

This one first!

Identifying Log Graphs

(6, 7)

(2, 0)

Type y = a log5 (x + b)

x y

x = 2, y = 0→ 0 = a log5 (2 + b)

must equal 1b = -1

y = a log5 (x – 1)

x y

x = 6, y = 7 7 = a log5 (6 – 1)

Identifying Log Graphs

Type y = a log5 (x + b)x = 2, y = 0

→ 0 = a log5 (2 + b)must equal 1

b = -1

y = a log5 (x – 1)x = 6, y = 7

7 = a log5 (6 – 1) 7 = a log5 (5) 1

7 = a y = 7 log5 (x – 1)

Identifying Log Graphs

(2, 3)

(-1, 0)

Type y = a log4 (x + b)

x y

x = -1, y = 0→ 0 = a log4 (-1 + b)

Identifying Log Graphs

Type y = a log4 (x + b)x = -1, y = 0

→ 0 = a log4 (-1 + b)must equal 1

b = 2

y = a log4 (x + 2)

x y

x = 2, y = 3 3 = a log4(2 + 2)

(2, 3)

(-1, 0) x y

Identifying Log Graphs

Type y = a log4 (x + b)x = -1, y = 0

→ 0 = a log4 (-1 + b)must equal 1

b = 2

y = a log4 (x + 2)x = 2, y = 3

3 = a log4 (2 + 2) 3 = a log4 (4) 1

3 = a y = 3 log4 (x + 2)

Identifying Log Graphs

Type y = a log5 (x + b)

Need to find a and b

mini log rules are vital again

This one first!

(6, 4)

(4, 0)

Type y = a log3 (x + b)

x y

x = 4, y = 0→ 0 = a log3 (4 + b)

must equal 1→ b = -3

y = a log3 (x – 3)

x y

x = 6, y = 4 4 = a log3 (6 – 3)

4 = a log3 3 1

Type y = a log3 (x + b)x = 4, y = 0

→ 0 = a log3 (4 + b)must equal 1

→ b = -3

y = a log3 (x – 3)x = 6, y = 4

4 = a log3 (6 – 3)

4 = a log3 3 1

4 = a y = 4 log3 (x – 3)

*

*

* Mini Log Rules

(6, 2)

(2, 0)

Type y = a log5 (x + b)

x y

x = 2, y = 0→ 0 = a log5 (2 + b)

must equal 1→ b = -1

y = a log5 (x – 1)

x y

x = 6, y = 2 2 = a log5 (6 – 1)

2 = a log5 5 1

2 = a y = 2 log5 (x – 1)

Key Question

(12, 15)

(5, 0)

Type y = a log2 (x + b)Nastier!

y = 5 log2 (x – 4)

(7, 8)

(-1, 0)

Type y = a log3 (x + b)

y = 4 log3 (x + 2)

1.

2.