Exponential and Logarithmic Functions

xby yx blogInverse

Exponent is another word for index.The variable x is the index (exponent)

Exponent is the logarithm

Base is always the base

Logarithms are useful in order to solve equations in which the unknown appears in the exponent

Reflection points, off y=x

422 Exponent is the logarithm.The output/y-value

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.

100010101010101010100010100 31212

1000log3 1010log1 10100log2 10

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.

Logarithms

http://www.sosmath.com/tables/logtable/logtable.html

A Table of the Common Logarithm

Common (or Briggian) logarithm (log) of the number. Base 10 logarithm

Natural (or Napierian) logarithms

3 is the common logarithm of 1000, since 103 = 1000. The base is 10. Exponent is the logarithm.

How many fingers do you have? What is the base of our number system? Our number system is based on powers of 10

Logarithmic tables have values between 1 and 10

Logarithmic Scale

0

200000

400000

600000

800000

1000000

1200000

Monday Tuesday Wednesday Thursday Friday

1

10

100

1000

10000

100000

1000000

Monday Tuesday Wednesday Thursday Friday

Amount of cars on I95 – Linear Scale Amount of cars on I95 – Logarithmic Scale

Hard to see anything here

All data is clearly seen

Day Vehicles %change

Monday 100  

Tuesday 10,000 9900.00%

Wednesday 100,100 901.00%

Thursday 50,000 -50.05%

Friday 1,000,000 1900.00%

Amount of cars on I95

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-to-Tuesday, not on Thursday-to-Friday.

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

ii aru

Continuous Exponential Growth/Decay

Initial value;Anything to the power 0 is a one

rate of growth (r>1)rate of decay (r<1)

number of time unites after start

Exponential Growth:•Rampant inflation•A nuclear chain reaction•Spread of an epidemic•Growth of cells

A geometric sequence with common ratio r.Functions having natural numbers.

discrete growth continuous growth

Functions having real numbers.

x = (0:.1:10)y = 10.*(.5.^x)plot (x,y)

xxxf )21(10)5(.10

Initial value

x = (0:.1:4)y = 10.*(2.^x)plot (x,y)

xxf )2(10

Graph never touches the x axis

Exponential Decay: radioactivity in lump of uranium ore, concentration of an antibiotic in the blood stream

Exponential Growth

Gets small in a hurry as x gets bigger. ½ When x is 1; 1024 when x is 10

)21(101 f

)10241(1010 f

Example

U.S. population in 1790: 3.9 million (initial value)

U.S. population in 1860: 31.4 million

U.S. population in 1990: ??? million

xbp 9.3

Population

Number of years

b = 1.030…

)17901990(1990 )030.1(9.3 p

)17901860()(9.34.31 b

70200

)17901990(

9.3

4.319.3)(9.3

bp

70

1

9.3

4.31

b

We could avoid solving for b

Decay of isotope carbon-14’s half life: 5715 years

By what percentage does carbon-14 decay in 100 years?

571515715 5.05.0 bb

0.5 units are left after 5715 years

100bx x units are left after 100 years

988.5.0 5715100

xWe lost .012 units

.012/1 = 0.012 = 1.2%

What percent of carbon-14 did we loose in 100 years – given it’s half life

Properties of Logarithms

qpq

p

qppq

xn

xx

xnx

nn

n

logloglog:ruleDivision

loglog)log(:ruletion multiplica

log1

loglog

loglog

:RulePower

1

1

22

1

222

2

21

2log2

1log1

2

12

2log4log242

2log122

01log b

nbnb log

10 b

2log3log32log3log2

3log5.13log

3log22log3log2log)32log(18log

204.10.301030042log42log16log

33

22

104

1010

Example:

Lookup log10 2 in the Table of the Common Logarithm.http://www.sosmath.com/tables/logtable/logtable.html

same

exponent is the logarithm

641.410

6666.03

2log

10log3

1log

100loglog

100100

6666.0

10

2

31

31

3

b

b

b

b

b

Example: Logarithmic Functions

481log3

1

31

181

481

1log3

81

14

13log813

481log381

31

4

4

34

814

1

34

In the old days, without calculators how could one find the cube root of 100 ?

reverse lookup (table of inverse function)

66666.010641.4 641.4log6666.0 10Inverse

All they did was one lookup and a simple division

do the log first and then undo the logdo the log first and then undo the log

5386.335386.0

10log456.3log10456.3log3456log 310

3

Please note this

How to find the logarithm of 3456 with a log table with values between 1 and 10

nbnb log

nbnb log

log of 4.64 is 0.6665180

exponent is the logarithm

24log42

12log22

22

21

x3log2

32 x

3log2log xTake the logarithm on both sides

3258.1301.0

477.0

2log

3log3log2log 58.1 xx

Power rule

Base 10

Convert to exponential form

Base 2 This is straightforward

Not straightforward

logb exponential form log10

unitst9174.0How many days does it take for the amount to fall less than 0.1 units?

unit1 t days

1.09174.0 t

Inequalities

1.0log9174.0log1.0log9174.0log tt

Take the logarithm on both sides

..708.269174.0log

1.0logt

switch the direction of inequality

a negative number

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

xxm 4)( xxq 2)( Grows faster)(2)( xqxk x

)(4)( xmxg x decays faster

xxxg

4

14)(

xxxk

2

12)(

Becomes clear as to why negative exponent is a decreasing function

, :Domain

,0 :Range All intercept here (0,1)Anything to the power 0 is a one

x axis is the horizontal asymptote

Graphs reflect on the y axis

Example 4 – page 301 -- Transformations

)1(3 1 xfxg x

Shift f(x) one unit to the left: add one to the input. Input is a bigger number takes off faster

2)(23 xfxh x

Shift f(x) down by 2 units: subtract 2 from the input

)(3 xfxk x

reflect f(x) on x axis: take the output and multiply it by -1

)(3 xfxi x

reflect f(x) on y axis: input is multiplied by -1 (negative input)

taby

Output is growing exponentially

a, b are constants

btabaaby tt loglog)log(log)log(log

Taking the logarithm of both sides, to any base

atby logloglog cmxy slope intercept

Graphs of Exponential Growth

Representing an exponential function as a linear function

Amount invested: $1000 Annual interest: 6%

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

One year interest = 1000 x .06 = $60

Amount at the end of the year = $1000 + $ 60 = $1060 = 1000 x 1.06

Geometric Series

nt

n

rA

1

number of compounds per year

Number of years

rtPeAFor large n; continuous compounding

1 xfxg

Shift f(x) one unit to the right

Example 6 (p314) Transformation of Graphs of Logarithmic Functions

xfxh 2Add two to the output

logarithm

x

xe

11....71828.2

31000log100010 103

exponent is the logarithm

nbnb

bbb

b

bn

b

b

b

loglog

1log

01log11

0

xn

xx

xnx

nn

n

log1

loglog

loglog1

qpq

p

qppq

logloglog

logloglog

3258.1301.0

477.0

2log

3log3log 58.1

10

102

changing base

Log functions and exponential functions are inverse of each other

Properties

transformations

nbnb

bbb

b

bn

b

b

b

loglog

1log

01log11

0

xn

xx

xnx

nn

n

log1

loglog

loglog1

qpq

p

qppq

logloglog

logloglog

xyxyya aaxa logloglog

Exponent is the logarithm

Bases are same

xa xa log

nb

nybbybn

b

nynb

log

log

base is same

3258.1301.0

477.0

2log

3log3log 58.1

10

102

31000log100010 103

Exponent is the logarithm

uncommon base

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.