solving exponential equations
TRANSCRIPT
Block 3
Solving Exponential Equations
What is to be learned?
• How to solve exponential equations
Exponential Equation?Equation where power is unknown
5x = 125 x = 3
3x = 20x = 2.73
(General Knowledge)
(Trial and error)
need a better tactic
The Log Tactic
8x = 900neat wee tactic
log 8 = log 900x = log 900
x
log 8
x = 3.27
log log
The Log Tactic
7x = 400neat wee tactic
x log 7 = log 400x = log 400
log 7
x = 3.08
log log
A bit nastier15 X 3x = 30003x = 3000
3x = 200 log 3x = log 200x log 3 = log 200 x = log 200
x = 4.82
15
log 3
A bit nastier20 X 4x = 80004x = 8000
4x = 400 log 4x = log 400x log 4 = log 400 x = log 400
x = 4.32
20
log 4
Solving Exponential Equations
Equations where unknown is a power
30 X 6x = 1200Eliminate 30
6x = 1200
6x = 40Logs on both Sides
Log 6x = log 40Neat wee tactic
x log 6 = log 40x = log 40
30
log 6= 2.06
(using log10)
Calculate x to 2 d.p.
80 X 7x = 47x = 4
7x = 0.05 log 7x = log 0.05x log 7 = log 0.05x = log 0.05
x = -1.54
80
log 7Key
Question
What is to be learned?
• How to solve exponential equations containing e
ex = 400Take Natural Logs of Both Sides (ln)
lnex = ln400Neat Wee Tactic
x lne = ln400
x = ln400 = 5.99
logee = 1
e0.4t = 80Take Natural Logs of Both Sides (ln)
lne0.4t = ln80Neat Wee Tactic
0.4t lne = ln80
0.4t = ln80 t = ln80
t = 10.96
logee = 1
0.4
e0.2t = 90Take Natural Logs of Both Sides (ln)
lne0.2t = ln90Neat Wee Tactic
0.2t lne = ln90 0.2t = ln90 t = ln90
t = 22.50
0.2
5e0.8t = 40 e0.8t = 8
Take Natural Logs of Both Sides (ln) lne0.8t = ln8
Neat Wee Tactic0.8t lne = ln8 0.8t = ln8 t = ln8
t = 2.60 0.8
Natural GrowthPopulation of some sort of creepy crawlieP = P0e0.2t
If there are 1000 ccs to start with, how long will it take to reach a million?1000000 = 1000e0.2t
1000000 = e0.2t
1000 = e0.2t
ln 1000 = ln e0.2t
ln 1000 = 0.2t ln e
P is Population over t days
1000
= 1
ln 1000 = 0.2t
t = ln 10000.2
= 34.5 days
P0
P
Solving Exponential Equations with e
Much better to use natural logs (ln) Remember lne = logee = 1
4e0.4t = 80Eliminate 4
e0.4t = 20Take Natural Logs of Both Sides (ln)
lne0.4t = ln20Neat Wee Tactic
0.4t lne = ln20 0.4t = ln20 t = ln20
t = 7.49
= 1
0.4
In Context ExampleTina’s tea temperature (T 0C) after t mins
T = T0e-0.2t
How long until it is half its initial temp?T = e-0.2t
0.5 = e-0.2t
ln 0.5 = ln e-0.2t
ln 0.5 = -0.2t ln e ln 0.5 = -0.2t
t = ln 0.5
T0
= 1
-0.2 = 3.5 mins
Theoretically Correct Approach
Could also choose values
for T and T0
2e0.8t = 450 e0.8t = 225
Take Natural Logs of Both Sides (ln) lne0.8t = ln225
Neat Wee Tactic0.8t lne = ln225 0.8t = ln225 t = ln225
t = 6.77 0.8 Key Question
Find t to 2 d.p.