the exponential & logarithmic functions exponential growth & decay worked example: joan puts...
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Higher Maths: Unit 3.3
The Exponential & Logarithmic Functions
Higher Maths: Unit 3.3Higher Maths: Unit 3.3Higher Maths: Unit 3.3Higher Maths: Unit 3.3Higher Maths: Unit 3.3Higher Maths: Unit 3.3Higher Maths: Unit 3.3Higher Maths: Unit 3.3Higher Maths: Unit 3.3
Exponential Growth & Decay
Worked Example:Joan puts £2500 into a savings account earning 13% interest per annum. How much money will she have if she leaves it there for 15 years?
Let £A(n) be the amount in her account after n years, then:
(0) 2500A 1(1) 2500 0.13 2500 2500 1.13A
2(2) (1) 0.13 (1) 2500 1.13A A A 3(3) (2) 0.13 (2) 2500 1.13A A A
( ) ( 1) 0.13 ( 1) 2500 1.13A n A n A n
nnnnnnn15(15) 2500 1.13 15635.68A
Example 1:The population of an urban district is decreasing at the rate of 2% per year.
(a) Taking P0 as the initial population, find a formula for the population, Pn, after n years.
(b) How long will it take for the population to drop from 900 000 to 800 000?
0 0 0(1) 0.02 0.98P P P P
1 1 1(2) 0.02 0.98P P P P 20 00.98 0.98 0.98P P
?
2 2 2(3) 0.02 0.98P P P P 2 30 00.98 0.98 0.98P P
Suggests that
00.98nnP P
(a)
Using our formula
00.98nnP P
And setting up the graphing calculator
In the fifth year the population drops below 800 000
Example 2:The rabbit population on an island increases by 15% each year. How many years will it take for the population to at least double?
0 0 0(1) 0.15 1.15P P P P
0PLet be the initial population
21 1 1 0(2) 0.15 1.15 1.15P P P P P
32 2 2 0(3) 0.15 1.15 1.15P P P P P
0( ) 1.15nP n P
Set 0 1P After 4 years the
population doubles
A Special Exponential Function – the “Number” e
The letter e represents the value 2.718….. (a never ending decimal). This number occurs often in nature
f(x) = 2.718..x = ex is called the exponential function to the base e.
Although we now think of logarithms as the exponents to which one must raise the base to get the required number, this is a modern way of thinking. In 1624 e almost made it into the mathematical literature, but not quite. In that year Briggs gave a numerical approximation to the base 10 logarithm of e but did not mention e itself in his work. Certainly by 1661 Huygens understood the relation between the rectangular hyperbola and the logarithm. He examined explicitly the relation between the area under the rectangular hyperbola yx = 1 and the logarithm. Of course, the number e is such that the area under the rectangular hyperbola from 1 to e is equal to 1. This is the property that makes e the base of natural logarithms, but this was not understood by mathematicians at this time, although they were slowly approaching such an understanding
Example 3:The mass of a fixed quantity of radioactive substance decays according to the formula m = 50e-0.02t, where m is the mass in grams and t is the time in years.
What is the mass after 12 years?
Linking the Exponential Function and the Logarithmic Function
y
x
( ) 2xf x
1( )f x (0,1)
(1,0)
In chapter 2.2 we found that the exponential function has an inverse function, called the logarithmic function.
log 1 0
log 1
log
a
a
xa
a
y a x y
2log x
The log function is the inverse of the exponential function, so it ‘undoes’ the exponential function:
f(x) = 2x
ask yourself:
1 2 21 = 2 so log22 = 1 “2 to what power gives 2?”
2 4 22 = 4 so log24 = “2 to what power gives 4?”
3 8 23 = 8 so log28 = “2 to what power gives 8?”
4 16 24 = 16 so log216 = “2 to what power gives 16?”
f(x) = log2x
23
4
Example 4:
(a)log381 = “…. to what power gives ….?”
(b)log42 = “…. to what power gives ….?”
1
27
(c)log3 = “…. to what power gives ….?”
4 3 81
4 2
-3 3
1
21
27
Rules of Logarithms
Rules for Logs:Rules for indices:
m n m na a a
m n m na a a
nm m na a
log log loga a axy x y
log log loga a a
xx y
y
log logpa ax p x
Example 5:
Simplify:
a) log102 + log10500 b) log363 – log37
10log (2 500)
10log 10003
Since310 1000
3
63log
7
3log 9
2
Since23 9
c) 2 2
1 1log 16 log 8
2 3
1
3
1
2
1
21
2
1
21
2 1
2
2 2log 16 log 8 1
3
1
2
1
3
1
31
3
1
3 3
2 2log 16 log 8 2 2log 4 log 2 2 1 1
Since2log 4 2
Since2log 2 1
Using your Calculator
You have 2 logarithm buttons on your calculator:
which stands for log10 and its inverse
which stands for loge and its inverse
log log
10x
ln
xe
ln
Try finding log10100on your calculator 2
Solving Exponential Equations
Solve 5 11x 51 = 5 and 52 = 25 so we can see that x lies between ……and…………..1 2
Taking logs of both sides and applying the rules
10 10log 5 log 11x xxxxxxxx 10 10log 5 log 11
10
10
log 111.489
log 5x
For the formula P(t) = 50e-2t:a) evaluate P(0)
b) for what value of t is P(t) = ½P(0)?
2 0(0) 50 50P e a)
b)1 1
(0) 50 252 2P
225 50 te21
2te
Could we have known this?
21ln ln
2te
21ln ln
2te
0.693 2 lnt e
0.693 2 1t 0.693
2t
0.346 t
The formula A = A0e-kt gives the amount of a radioactive substance after time t minutes. After 4 minutes 50g is reduced to 45g.(a) Find the value of k to two significant figures.
(b) How long does it take for the substance to reduce to half it original weight?
Example
(a) 4t (0) 50A (4) 45A
445 50 keTake logs of
both sides 4ln(45) ln 50 ke 4ln(45) ln 50 ln ke 4ln 45 ln 50 ln ke
4ln 45 ln 50 ln ke 45
ln 4 ln50
k e
0.1054 4k remember
ln 1e
0.1054
4k
0.0263 k
(b) How long does it take for the substance to reduce to half it original weight?
0.02631(0) (0)
2tA A e
0.02631
2te
0.02631ln ln
2te
0.693 0.0263 lnt e
0.693 0.0263 26.35t t
Experiment and Theory
When conducting an experiment scientists may analyse the data to find if a formula connecting the variables exists. Data from an experiment may result in a graph of the form shown in the diagram, indicating exponential growth. A graph such as this implies a formula of the type y = kxn
2 2 4 6 8 10
y
x
We can find this formula by using logarithms:
ny kxIf
Then log log ny kx
So log log ny kx log k log nx
log log ny kx log k logn xCompare this to Y mX c
log y Y m n logc k
So log log ny kx log k logn xIs the equation of a straight line
log y
log x
log y
log x
From ny kx
We see by taking logs that we can reduce this problem to a straight line problem where:
So log log ny kx log k logn x
Andlog y Y m n logc k
Worked Example:
The following data was collected during an experiment:
x 50.1 194.9 501.2 707.9
y 20.9 46.8 83.2 102.3
a) Show that y and x are related by the formula y = kxn
.
b) Find the values of k and n and state the formula that
connects x and y.
a) Taking logs of x and y gives:
logx
logy
1.69 2.28 2.70 2.841.32 1.67 1.92 2.00
Plotting these points
0.5
0.5
1
1
1.5
1.5
2
2
2.5
2.5
3
3
0.5
0.5
1
1
1.5
1.5
2
2
2.5
2.5
3
3
log10 Y
log10 X
We get a straight line and hence the formula connecting X and Y is of the form
y mx c
b)Since the points lie on a straight line, we can say that:
ny kxIf
Then log log ny kx
So log log ny kx log k log nx
log log ny kx log k logn xCompare this to Y mX c
By selecting points on the graph and substituting into this equation we get using
1.69,1.32 2.84,2.00
1.32 1.69m c 2.00 2.84m c
Subtract
0.68 1.15m0.68
1.15m
0.6 m
1.32 1.69 0.6 c
So
0.3 c
So we have 0.6 0.3Y X Compare this to
log log ny kx log k logn x
n and log k 0.6 0.3so
solving
0.310k
log k 0.3
1.99k
so 0.61.99ny kx y x
You can always check this on your graphics calculator