Block 3

Finding Exponential Equations

R

t

xxx

xx

x

xNot very straight!

t

xxx

xx

x

x

Exponential

Equation?

R

t

Rlog

log

xx

x

xx

xx

find equation with log R and log tin form log R = m log t + c

t

Rlog

log

xx

x

xx

xx

need two points on linegradient

6

(4 , 34)m = 34 – 6

4 – 0 = 28/4

= 7

y = 7x + 6

log R = 7 log t + 6

do not want logs in equation

log R = 7 log t + 6

logless!

want to change 6 → log10........

log statement log10 x = 6

translation x = 106

equation becomeslog R = 7 log t +

still some work to do!no need to always write 10

?

log10 106

log R = 7 log t + log106

to get rid of logs want log…. = log….

Time for some log rules

log R = 7 log t + log106

Neat Wee Rulelog R = log t7 + log106

Addition Rulelog R = log (t7 X 106)

R = 1000000t7

TacticsGet straight line equation with logsChange everything to logsGet rid of the logs!!

Finding Exponential EquationsUsing Logs

t

Rlog

log

xx

x

xx

xx

find equation with log R and log tin form log R = m log t + c

type y = axn

t

Rlog

log

xx

x

xx

xx

need two points on lineFind Gradient

8

(13 , 34)m = 34 – 8

13 – 0 = 26/13

= 2

y = 2x + 8

log R = 2 log t + 8

log R = 2 log t + 8

logless!

want to change 8 → log10?

log statement log10 x = 8

translation x = 108

equation becomeslog R = 2 log t + log 108

no need to always write 10

log R = 2 log t + log108

Neat Wee Rulelog R = log t2 + log108

Addition Rulelog R = log (t2 X 108)

R = 100000000t2

to get rid of logs want log…. = log….

Eliminate LogsR = (t2 X 108)

m = 2 , y int = 4→ y = 2x + 4→ log H = 2log M + 4

log H = 2log M + log104

log H = log M2 + log 10000log H = log 10000M2

H = 10000M2

log H

log M

4(4 , 12)

Key Question

t

Rlog

xx

x

xx

xx

Equation log R = mt + c

t

Rlog

xx

x

xx

xx

4

(6 , 7)m = 7 – 4

6 – 0 = 3/6

= 0.5

y = 0.5x + 4

log R = 0.5t + 4

log R = 0.5t + 4

logless!

want to change 4 → log10

log statement log10 x = 4

translation x = 104

equation becomeslog R = log 100.5t + log 104

same for 0.5t

log10 x = 0.5t

x = 100.5t

log R = log 100.5t + log 104

Addition Rule

log R = log (100.5t X 104)

R = 10000 X 100.5t

Eliminate LogsR = (100.5t X 104)

100.5 = 3.2R = 10000 X 3.2t

R = 10000 (3.2)t

m = 0.25 , y int = 3→ y = 0.25x + 3→ log R = 0.25G + 3

Same tactics as previouslylog R = log 100.25G + log 103

Log Rulelog R = log (100.25G X 103)

Remove LogsR = (100.25G X 103) = (103 X 100.25G) = (103 X 1.78G)

R = 1000(1.78)G

log R

G

3(11 , 5)

Type y = abx

Both logless!

(100.25 = 1.78)

m = 0.2 , y int = 1→ y = 0.2x + 1→ log F = 0.2P + 1

log F = log 100.2P + log 101

log F = log (100.2P X 10)F = (100.2P X 10) = (10 X 100.2P) = (10 X 1.58P)

R = 10(1.58)P

log F

P

1(8 , 3)

Type y = abx

Key Question

If y = abt

Show that the relationship between log y and log b is linear

log y = log abt

reverse addition rulelog y = log a + log bt

Neat Wee Rule log y = log a + t log b

log y = t log b + log a

eh?log both sides

Annoying First Part of Question Sort of Thing

like y = mx + c → linear

i.e in the form y = mx + c