Download Exponential equations and logarithms

Exponential equations and
logarithms
Fun with exponential equations
 Every pair share a packet of M&Ms.
 You are supposed to firstly empty the M&Ms on the serviettes


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
provided randomly.
Count the number of M&Ms in all and record your reading as the
zeroth reading. Then eat all the M&Ms facing up (i.e. with the
letters on top) and count those remaining as the first reading.
Now put the M&Ms back into the bag, shake and pour out again.
Eat those facing up and count the remaining number of M&Ms
again and record your reading as the second reading.
Continue the procedure until you have no more M&Ms remaining.
Fun with exponential equations
 Now plot these points on Microsoft Excel, following my
instructions.
 What kind of graph does your readings give?
 The functions that gives this kind of graphs are called
exponential functions.
Exponential equations
 Exponential Growth
 Exponential Decay
The Beggar and the King
The Beggar and the King
 Do you think the beggar made a wise request?
 Was the King right in agreeing to the beggar’s request?
 What do you think will happen?
The Beggar and the King
Day
Weight of rice/ g
1
2
2
2x2=22=4
3
2x2x2=23=8
4
2x2x2x2=24=16
5
2x2x2x2x2=25=32
6
2x2x2x2x2x2=26=64
7
2x2x2x2x2x2x2=27=128
Exponential Growth
The Beggar and the King
 Moral of the story:
 Know your Mathematics! Use them in everyday life!
 学以致用！
Folding a piece of paper
 http://raju.varghese.org/articles/powers2.html
Exponential Decay of radioactive
substances
 Cesium-137 and strontium-90 present long-term
environmental hazards and can be absorbed throughout the
body, particularly bones. Plutonium-239 exposure often
leads to lung cancer, and it has a half-life of 24,000 years, so
it would be around for a long, long time.
 (A half-life is the amount of time it takes for half of the
radioactive isotopes in a substance to decay.)
Exponential decay
Graphs of exponential functions
y = 3x
y = 2x
Graphs of exponential functions
1
y  ( )x
3
1
y  ( )x
2
Graphs of exponential functions
y  2 x
y  2x
Looks the same? Why?
1
y  ( )x
2
Logarithm
Index
Index
form
ya
log a y  x Logarithmic
form
x
Base
For any positive number a, except 1, y  a
x
 log a y  x
What you want to know is: a to the power of WHAT gives y?
This WHAT is your x, which is what you want to find in the
logarithmic form.
Logarithm
For any positive number a, except 1, y  a
x
 log a y  x
 Why can’t a =1 or a < 0?
 a is the base of the logarithm. It cannot be equal to 1 as 1x is
always 1. a cannot be negative as powers of negative numbers
change sign.
 For logay to be defined:
 y > 0, why?
 a > 0, a ≠ 1
Logarithm
 For logay to be defined:
y>0
 a > 0, a ≠ 1
 The following are not defined:
log12
2. log-34
3. log2(-1)
1.
Logarithm
Convert to logarithmic form.
What is the base here?
1
What is the index here?
4 
64
1
log 4
 3
64
3
Convert to index form.
What is the base here?
log 6 36  2
6 2  36
What is the index here?
Convert the following to logarithmic
form
16  4 2
103  1000
1
1
2 
2
y  a  log a y  x
x
Convert the following to index form
3  log 2 8
2  log 5 25
1
 log 2 2
2
Solve the following equations
2  log 3 x
x  log 1 9
3
2  log x (4 x  3)
log x 4  2
2 important conclusion
a  a1  log a a  1
a 0  1  log a 1  0
log a a  1
log a 1  0
What conditions of a do you need to impose here?
Evaluate the following
log 5 (2 log 6 6  log 3 3)
The Common Logarithm
 Logarithms with a base of 10 are called common logarithms
 Use the LOG key on your calculator to find the values of
common logarithms.
 log10x is often abbreviated as lgx.
 Uses: In chemistry as a measure of acidity, in earthquakes as a
measure of the strength (Richter Scale) etc. For earthquakes,
the wave amplitude is typically very big, so a common
logarithm scale is used. Think: a 9.0 earthquake (Japan) is
how many times more powerful (in terms of wave
amplitude) than an 8.0 earthquake (Sichuan earthquake)?
Solve the following equations, using
your calculators, giving your answers
correct to 4 s.f.
lg( 3 x  2)  1.24
7
x 1
 99.5
x = 1.364
x
0.5(5 )  34(5 )  12
x
x = 6.459
x = 1.883 or x=0.7384
Natural Logarithm
 There is another logarithm to the base of a special irrational





number called e, named after Leonhard Euler.
e has a value of 2.7183……….
Logarithm to the base e, logex, is often abbreviated as ln x.
ln x is called Natural logarithm, or Naperian logarithm (after
John Napier)
Can you find e and ln on your calculator?
Where is it used? Radioactive decay, first order reaction in
Chemistry, calculus etc.
Solve the following equations
e
2 x 1
7
e 2 x  5e x  6
Laws of logarithm
log a mn  log a m  log a n
Product Law
m
log a  log a m  log a n
n
Quotient Law
log a m  k log a m
Power Law
k
Note: m and n are positive and a > 0, a ≠ 1
Proofs
 Product Law:
Let log a m  x and log a n  y
m  a ;n  a
x
y
m n  a  a
x
x y
y
mn  a
 log a mn  x  y  log a m  log a n
Proofs
 Quotient Law:
Let log a m  x and log a n  y
m  a ;n  a
x
y
mn  a a
x
y
m
x y
a
n
m
 log a  x  y  log a m  log a n
n
Proofs
 Power Law:
Let log a m  x
ma
x
m  (a )  a
k
x k
xk
 log a m  xk  k log a m
k
Are the following true?
log a ( x  y)  log a x  log a y
log a ( x  y)  log a x  log a y
log a x  (log a x)
r
r
Simplify the following
log 3 5  log 3 7  3 log 3 2
log 4 2  log 4 8
2
81
3
2 log 2  log 2  2 log 2
3
8
4
4
log 10  log 10 70  log 10 2  2 log 10 5
35
Logarithmic equations of the same
base
 For two logarithms of the same base,
log a M  log a N  M  N
Solve the following equation
log 10 (3x  2)  2 log 10 x  1  log 10 (5x  3)
5x  x  6  0
(5 x  6)( x  1)  0
6
x   or x  1
5
Reject any of them?
2
Solve the following equation
log 2 ( x 1)  log 2 ( x  4)  log 2 (2x  6)
x  5x  4  2 x  6
2
x  7 x  10  0
2
( x  2)( x  5)  0
x  2 or x  5
Reject any of them?
Solve the following simultaneous
equations
log 2 ( x  4 y )  4
log 8 4 x  log 8 (8 y  5)  1
x  18
1
y
2
Change of bases
 If a, b and c are positive numbers and a≠1,c≠1, then:
log c b
log a b 
log c a
Proof
Let x  log a b, then a x  b
Take logarithms , base c, of both sides :
log c a x  log c b
x log c a  log c b
log c b
x
log c a
log c b
log a b 
log c a
log b b
1
When c  b, log a b 

,
log b a log b a
so log a b 
1
log b a
Example
 Evaluate log75 × log59 × log97
 Hint: Change all the bases to common log!
lg 5
log 7 5 
lg 7
lg 9
log 5 9 
lg 5
lg 7
log 9 7 
lg 9
lg 5 lg 9 lg 7
log 7 5  log 5 9  log 9 7 


1
lg 7 lg 5 lg 9
Example
 Evaluate
log 5 4  log 2 10
log 25 10
Hint: Change all the bases to common log!
Ans: 8
Solve the following equation
log 5 x  4 log x 5  3
log 4 (6  x)  log 2 8  log 9 3
Examples (using all you have learnt)
Solve log 4 (6  x)  log 2 8  log 9 3.
Given that log43 = a and log45 =b, express log445 in terms of
a and b.
Exercise
 Exercise 2.4
 Qns 4e-h, 5d-f, 6, 7, 8, 9, 10
 Exercise 2.5
 Qns 2e-f, 3e-f, 4e-f
 Exercise 2.6
 Qns 4c, 5d-f, 6a-d, 7, 8, 9, 10d-f, 11
 Exercise 2.7
 Qns 1e-h, 2, 3, 4a, 5c-e, 6c-d, 7, 8
Equations of the form ax=b
 Remember these?
1
1. 4 
16
 Can you solve the equations
below like what you did?
x
2. 52 x 1  1
3. 92 x1  27 x  0
1
1. 3 
16
x
2. 52 x1  2
3. 92 x1  5 x  0
Equations of the form ax=b
 If you cannot express both sides of the equation with the
same base, the strategy is to take LOGARITHM on both
sides.
Solve the following equations:
6 x  2  21
10 2 x 3  0.5
e3 x  9
Example
Express 3x (22 x )  7(5 x ) in the form a x  b. Hence, find x.
Hint : Combine all those with the same power toge ther!
x = 2.22
Graphs of y=e-x and y=ex
y  e x
y  ex
Graphs of lgx and lnx
y  ln x
y  lg x
Exercise 2.8
 Qns 1h-i, 3, 4b-c, 5b, 5d, 6b, 6c, 7a-b, 8