Transcript Objectives - College of Southern Maryland

Exponential and Logarithmic Functions
Logarithms are useful in order to solve equations
in which the unknown appears in the exponent
Exponent is another word for index.
The variable x is the index (exponent)
Exponent is the logarithm
y b
x
Inverse
x  logb y
Base is always the base
Reflection points, off y=x
22  4
Exponent is
the logarithm.
The output/yvalue
Corresponding
input/x-value
Input 4 produces a smaller output 2  a log function
grows slower than an exponential function
Objectives
• Understand the idea of continuous exponential growth and
decay
• Know the principal features of exponential functions and
their graphs
• Know the definition and properties of logarithmic functions
• Be able to switch between the exponential and logarithmic
forms of an equation
• Understand the idea and possible uses of a logarithmic
scale
• Be familiar with the logarithms to the special base e and 10
• Be able to solve equations and inequalities with the
unknown in the index
• Be ale to use logarithms to identify models of the form
y = abx and y = axn.
Logarithms
Common (or Briggian)
logarithm (log) of the
number. Base 10 logarithm
3 is the common logarithm of
1000, since 103 = 1000. The
base is 10. Exponent is the
logarithm.
Natural (or Napierian)
logarithms
How many fingers do you have?
What is the base of our number
system? Our number system is
based on powers of 10
2  log10 100
Logarithmic tables have
values between 1 and 10
1  log10 10
3  log10 1000
10010  1000 101010  102 101  1021  103  1000
Adding 2+1 is easier than multiplying 100 x 100 
this is why the logarithms were invented in the 17th
century  multiplication by addition and division
by subtraction.
A Table of the Common Logarithm
http://www.sosmath.com/tables/logtable/logtable.html
Logarithmic Scale
Amount of cars on I95
Day
Vehicles
%change
Monday
100
Tuesday
10,000
9900.00%
100,100
901.00%
50,000
-50.05%
1,000,000
1900.00%
Wednesday
Thursday
Friday
Percentage wise, 100 to 10,000 is much
larger increase than 50,000 to 1000,000
The highest 9900% increase
in traffic is not clear at all on
a linear scale. One would
wrongly conclude that the
largest day-to-day percent
increase happened on Friday
The biggest difference in
the daily graph heights
occurred on Monday-toTuesday, not on
Thursday-to-Friday.
1200000
1000000
1000000
100000
800000
10000
600000
1000
400000
100
200000
10
0
Logarithmic scale is suitable
for large data swings, such
as here, the number of cars
on the highway goes from
100 to 1,000,000
1
Monday
Tuesday
Wednesday
Thursday
Amount of cars on I95 – Linear Scale
Friday
Monday
Tuesday
Wednesday
Thursday
Amount of cars on I95 – Logarithmic Scale
All data is clearly seen
Hard to see anything here
Friday
Continuous Exponential Growth/Decay
f x   10(.5)  10( 1 ) x
2
f 1  10( 1 )
2
x
number of time
unites after start
f 10  10( 1
)
1024
Gets small in a hurry
as x gets bigger.
½ When x is 1; 1024
when x is 10
Initial value
x = (0:.1:10)
y = 10.*(.5.^x)
plot (x,y)
ui  ar
i
Initial value;
Anything to the
power 0 is a one
Graph never
touches the x
axis
rate of growth (r>1)
rate of decay (r<1)
discrete growth
A geometric sequence
with common ratio r.
Functions having natural
numbers.
Exponential Growth:
•Rampant inflation
•A nuclear chain reaction
•Spread of an epidemic
•Growth of cells
continuous growth
Functions having real
numbers.
Exponential Decay: radioactivity in lump of uranium
ore, concentration of an antibiotic in the blood stream
f x  10(2) x
x = (0:.1:4)
y = 10.*(2.^x)
plot (x,y)
Exponential Growth
Example
U.S. population in
1790: 3.9 million
(initial value)
U.S. population in
1990: ??? million
Number of years
p1990  3.9(1.030)(19901790)
p  3.9b x
Population
31.4  3.9(b)
(1860 1790 )
U.S. population in
1860: 31.4 million
 31.4 
b

3
.
9


We could avoid
solving for b
p  3.9(b)
(1990 1790 )
 31.4 
 3.9

 3.9 
200
70
1
70
 b = 1.030…
What percent of carbon-14 did we loose in 100 years – given it’s half life
Decay of isotope carbon-14’s
half life: 5715 years
0.5  b
0.5 units are left
after 5715 years
5715
By what percentage does carbon14 decay in 100 years?
 b  0.5
1
5715
x  b100
x units are left
after 100 years
x  0.5
100
5715
 .988
We lost .012 units
.012/1 = 0.012 = 1.2%
Properties of Logarithms
logb 1  0
b0  1
21  2  1  log2 2
 
2 2  4  2  log2 4  log2 2 2
2 1 
 
1
1
 1  log    log2 2 1
2
2 2 
Power Rule :
exponent is the logarithm
logb bn  n
Lookup log10 2 in the Table of the Common Logarithm.
http://www.sosmath.com/tables/logtable/logtable.html
Example:
log x n  n log x
same
log10 16  log10 2 4  4 log10 2  40.3010300  1.204
1
log18  log(2  32 )  log 2  log 32  log 2  2 log 3
log x  log x  log x
n
33
log13.5  log  log 33  log 2  3 log 3  log 2
multiplication rule : log( pq)  log p  log q
2
p
Division rule : log  log p  log q
q
n
1
n
Example: Logarithmic Functions
81  34  log3 81  4
How to find the logarithm of 3456 with a log table with values
between 1 and 10
1
3  81  log81 3 
4
1
1
 3 4  log3  4
81
81
1
4
1
81 
1
3


log3456 log 3.456103  log3.456 log10 103
 0.5386 3  3.5386
4
 
4
1
    log 1 81  4
3
3
Please note this
logb bn  n
In the old days, without calculators how could one find the cube root of 100 ?
1
b  100  100 3
3
1
log b  log100 3
1
log b  log102
3
log b  n
2
log10 b   0.6666
3
 b  100.6666  4.641
All they did was one lookup
and a simple division
log of 4.64 is 0.6665180
reverse lookup (table
of inverse function)
n
b
exponent is the
logarithm
do the log first and then undo the log
4.641 100.66666
Inverse
0.6666 log10 4.641
logb  exponential form  log10
Base 2
log2 3  x
This is straightforward
21  2  log 2 2  1
Not straightforward
2 2  4  log 2 4  2
Convert to exponential form
2x  3
Take the logarithm on both sides
Base 10
log 2 x  log3
Power rule
log3 0.477
x log 2  log3  x 

 1.58  21.58  3
log 2 0.301
Inequalities
1 unit
t days
t
0.9174 units
How many days does it take for the
amount to fall less than 0.1 units?
0.9174t  0.1
Take the logarithm on both sides


log 0.9174t  log0.1  t log0.9174 log0.1
a negative number
t
log 0.1
 26.708..
log 0.9174
switch the direction of inequality
The iodine-131 will fall to less than 0.1 units after about 26.7 days.
Example 2, 3 – page 299 -- ax and a-x
Graphs reflect
on the y axis
decays faster
x
k ( x)  2 x
g ( x)  4
 q( x)
 m( x)
Grows faster
x
m( x)  4
q( x)  2x
g ( x)  4  x 
k ( x)  2  x 
Range : 0, 
1
4x
1
2x
Becomes clear as to
why negative exponent
is a decreasing function
All intercept here (0,1)
Anything to the power 0 is a one
Domain:  , 
x axis is the horizontal asymptote
Example 4 – page 301 -- Transformations
Shift f(x) one unit to the left: add one to the input.
Input is a bigger number  takes off faster
g x  3x1  f ( x  1)
Shift f(x) down by 2 units: subtract 2 from the input
hx  3x  2  f ( x)  2
reflect f(x) on x axis: take the output and multiply it by -1
k x  3x   f ( x)
reflect f(x) on y axis: input is multiplied by -1 (negative input)
ix  3 x  f ( x)
Graphs of Exponential Growth
Output is growing exponentially
y  abt
a, b are constants
Representing an
exponential
function as a
linear function
Taking the logarithm of
both sides, to any base
log y  log(abt )  log a  log(bt )  log a  t logb
log y  logbt  log a
slope
intercept
y  mx  c
Amount invested: $1000
Annual interest: 6%
One year interest = 1000 x .06 = $60
Amount at the end of the year = $1000 + $ 60 = $1060 = 1000 x 1.06
Amount after 1 year
$1000 x (1+ 06) = $1060 = 1000 x 1.061
Amount after 2 years
$1060 x 1.06 = $1123.6 = 1000 x 1.062
Amount after 3 years
$1124 x 1.06 = $1191 = 1000 x 1.063
number of
compounds
per year
 r
A  1  
 n
Number of years
nt
For large n; continuous compounding
A  Pert
Geometric Series
Example 6 (p314)
Transformation of Graphs of Logarithmic Functions
Add two to the output
hx  2  f x
g x  f x 1
Shift f(x) one unit to the right
b 0  1  logb 1  0
b  b  logb b  1
1
103  1000 log10 1000 3
logb b n  n logb b  n
exponent is the logarithm
log x  n log x
n
1
log x  log x  log x
n
log pq  log p  log q
n
log
1
n
Properties
logarithm
changing
base
log2 3 
log10 3 0.477

 1.58  21.58  3
log10 2 0.301
p
 log p  log q
q
 1
e  2.71828....  1  
 x
x 
Log functions and exponential functions are inverse of each other
transformations
b 0  1  logb 1  0
logb b n  y  b y  b n  y  n
b1  b  logb b  1
 logb b n  n
logb b n  n logb b  n
base is same
log x  n log x
n
1
log x
n
log pq  log p  log q
log n x  log x
log
1
n

uncommon base
log2 3 
p
 log p  log q
q
log10 3 0.477

 1.58  21.58  3
log10 2 0.301
Exponent is the logarithm
aloga x  y  loga y  loga x  y  x
Bases are same
a loga x  x
Exponent is the logarithm
103  1000 log10 1000 3
Excel has built-in functions to calculate the logarithm of a number with a
specified base, the logarithm with base 10, and the natural logarithm.