Topic 6 Log Edited

7/27/2019 Topic 6 Log Edited

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X INTRODUCTIONBoth exponential and logarithm functions are one-to-one functions. The inversefunction of an exponential function is called the logarithm function. This topicwill discuss how these two functions associate with one another.

PROPERTIES OF EXPONENTIALS6.1

TTooppiicc

66X

ExponentialandLogarithmFunctions

LEARNING OUTCOMES

By the end of this topic, you should be able to:

1. Calculate equations involving exponentials and logarithms;

2. Sketch graphs of exponential and logarithm functions; and

3. Solve problems using the applications of the two functions.

Between linear and exponential functions, which one has the most rapidchange in its values?

ACTIVITY 6.1


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X TOPIC 6 EXPONENTIAL AND LOGARITHM FUNCTIONS96

A function fis called an exponential function if it has a form f(x) = ax where the

base ais positive, with a 0 and that its exponent xis any real number.

Examples:Find the values of

(a) 32 33 (b) (22)3 (c)3

24

(d) 32 (e)3

1

2

(f)2

3

2

Solutions:(a)

2 3

2 3

1

3 3

3

3

3

+

=

=

=

(b) ( )3

2 2 3

6

2 2

2

64

=

=

=

(c) ( )3

32

3

4 4

2

8

=

=

=

(d) 22

13

3

1

9

=

=

(e) ( )3

31

3

12

2

2

8

=

=

=

(f)2 2

2

3 3

2 2

1 1

9 4

4

9

=

=

=

(1) axay= ax+y(2) axbx= (ab)x(3) x x y

y

aa

a

=

(4) xxx

a a

b b

=

(5) (ax)y= axy(6) 1x

xa

a

=

(7) a0= 1(8) a1= a(9) ( )x xyya a=


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TOPIC 6 EXPONENTIAL AND LOGARITHM FUNCTIONS W 97

Examples:Solve

(a) 82x= 2 (b) 2 1 1xee

+ =

(c) 11

2 28

x x = (d) 3x2 94x= 0

Solutions:(a)

( )

2

23

6 1

8 2

2 2 (Equate the base)

2 2 (Compare the exponent)

6 1

1

6

x

x

x

x

x

=

=

==

=

(b) 2 1

2 1 1

1

2 1 1

2 2

1

x

x

ee

e ex

x

x

+

+

=

=

+ =

=

=

(c) 1

1 3

12 2

8

2 2

2 1 3

2 2

1

x x

x x

x

x

x

+

=

=

=

=

=

(d)

( )

( )( )

2

2

4

42

2

2

3 9 0

3 3

8 2

2 8 0

2 4 0

2, 4

x x

xx

x x

x x

x x

x x

=

=

=

+ = + =

= =

Find the values of

(a) 3 34 (b) 2-3 8 (c)1

327

(d)

2

31

8

(e)3

1

5

(f) 42 21

EXERCISE 6.1


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EQUATIONS AND GRAPHS SKETCHING

There are two general shapes of exponentials graphs. The shapes depend on thebase value of the exponential functions.

(a) y= axwhere a> 1

Graph 6.1(b) y= axwhere 0 < a< 1

Graph 6.2

6.2

Solve

(a) 1 164

x

=

(b) 3 1xe + = (c) 4x 2x+1 = 0

(d) 2x8x= 2 (e) (f) ( )( )2 2 1x x

e ee

= (f) ( )( )2 2 1x x

e ee

=

EXERCISE 6.2


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The followings are the properties of the graph of exponential function f(x) = ax.

(i) The domain for exponential functions is the entire real numbers.(ii)

Its range is all positive numbers.

(iii) They-intercept on the exponential graph is (0,1).(iv) There is no x-intercept.(v) Ifa> 1, the graph is increasing form left to right.(vi) If 0 < a< 1, the graph is decreasing form left to right.Example:Sketch a graph ofy= 2x.

Solution:(i) Construct a table consisting several values ofxandy.(ii) Plot the points on a plane.(iii) Draw a smooth curve through all the plotted points.(i)

x 2 1 0 1 2 3

y 14

12

1 2 4 8

(ii)

Graph 6.3


6/18


Example:Sketch a graph of

1

2

x

y

=

.

Solution:(i)

x 3 2 1 0 1 2

y 8 4 2 11

2

1

4

(ii)

Graph 6.4

LOGARITHM FUNCTIONS

A logarithm function with base a,is written as log a where a> 0, a 1. yis thelogarithm for xwith base a,denoted by y= log ax.

6.3

ay =x

Exponential Form

y = loga x

Logarithm Form

Is logarithm function a reciprocaloperation for exponential function?

ACTIVITY 6.2


7/18


Example:Convert the following equations, from logarithm to exponential forms.

(a) log39 = 2(b) log10 y= 4

(c) log28 = 3

Solution:(a) 32 = 9

(b) 104=y

(c) 23 = 8

Example:Convert the following equations, from exponential to logarithm forms.

(a) 25 = 32(b) 100= 1(c) 53 =y

Solution:(a) log 232 = 5(b) log 101 = 0(c) log 5 y= 3

Logarithm with base of 10 is known as common logarithm, and is written as log

10 x= logx= lgx. Logarithm with base e, is called natural logarithm, denoted bylog ex= lnx.

PROPERTIES OF LOGARITHMS6.4

What is the value for e? What is the significance ofe?

ACTIVITY 6.3


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Examples:Using the above properties, find the value for:

(a) log 381 (b)1

lne

(c) log a1

(d) log 42 (e) log 42 + log 48 (f) log 654 log 69

Solutions:(a)

( )

4

3 3

3

log 81 log 3

4 log 3

4 1

4

=

==

=

(b)

( )

1

1 1ln log

log

1log

1 1

1

e

e

e

e e

e

e

=

=

=

=

=

(c)

( )

0log 1 log

0 log

0

a a

a

a

a

=

==

(d)4 4

1

24

4

log 2 log 4

log 4

1log 4

2

1

2

=

=

=

=

(e)

( )

4 4 4

2

4

4

log 2 log 8 log 16

log 4

2log 4

2 1

2

+ =

=

=

=

=

(f) 6 6 6

6

54log 54 log 9 log

9

log 6

1

=

=

=

(1) log aa= 1(2) log amx= xlog am(3) log am= log

log

b

b

m

a(Logarithm base interchangeable formula)

(4) log aM+ log aN= log aMN(5) log log loga a a MM N

N =

(6) log aM =log aN then M= N


9/18


Examples:Find the value ofx.

(a) log (2x+ 1) = log (x+ 6) (b) logx(6

x) = 2(c) log3x= 2 (d) log x= 1(e) log 2x4 + log 2 4x= 12 (f) log x log (x 1) = log 4

Solutions:Find the value ofx.

(a) ( ) ( )log 2 1 log 6

2 1 6

2 6 1

5

x x

x x

x x

x

+ = +

+ = +

=

=

(b) ( )

( )( )

2

2

log 6 2

6

6 03 2 0

3, 2

3will be ignored as the base 0

So, 2

xx

x x

x xx x

x x

x x

x

=

=

+ =+ =

= =

= >

=

(c)3

2

log 2

3

9

x

x

x

=

=

=

(d)

10

1

log 1

log 1

10

x

x

x

=

=

=

(e)

( )

( )

4

2 2

4

2

5

2

5 12

125

2

5 10

1

10 5

2

log log 4 12

log 4 12

log 4 12

4 2

2

2

2

2

2

4

x x

x x

x

x

x

x

x

x

x

+ =

=

=

=

=

=

=

=

=

(f) ( )log log 1 log 4

log log 41

41

4 4

3 4

4

3

4

3

x x

x

x

x

x

x x

x

x

x

=

=

=

=

=

=

=


10/18


11/18


12/18


Solution:(a) Convert the equation, from logarithm to exponential form.

(b) Construct a table consisting several values ofxandy.(a) Draw a smooth curve through all the points.(a) y= log2 x

2y= x

(b)

y 2 1 0 1 2 3

x

1

4

1

2 1 2 4 8

(c)

Graph 6.7Example:Sketch a graph ofy= log 1/2x

Solution:(a)

1

2

y

x

=

(b)

y 3 2 1 0 1 2

x 8 4 2 11

2

1

4


13/18


(c)

Graph 6.86.5.1 Application on Growth and Decay ProcessesExponential functions can be applied into growth and decay processes. Theformula for total growth is

Where

P =number of residents after tyears.

P0 =number of original residents.

r =percentage (rate) of growth

t =time period

Example:Suppose the total number of residents in a given town is 20,000 and the rate ofgrowth of the residents is 5% per year.

(a) Determine the total number of residents in this town in the period of 6years from now.

(b) How many years will it take for the number of residents to double?

P = P0 ert


14/18


Solution:(a) Substitute all the given values into the formula to find the value ofP.

P = P0ert, where P0= 20,000, r = 5% and t= 6.= 5/100= 0.05

P=20000e0.05(6)= 20000e0.3= 26997

Hence, the number of the town residents after six more years is 26997.

(b) Doubling the number of residents implies P=2Po.

Substituting Pwith2Po and r= 0.05 into the formula to find the value for t.

o

0.05

o o

0.05o

o

0.05

2

2

2

log 2 0.05

ln 2 0.05

ln 2

0.05

13.863

rt

t

t

t

e

P P e

P P e

Pe

P

e

t

t

t

t

=

=

=

=

=

=

=

=

The numbers of residents will double in about 14 years.

The formula for decay process is

Example:Suppose a radioactive element is going through power decay after tdays based

on exponential function P=100 e0.075t. How much of the quantity is left after 20days?

P = P0 e

-rt


15/18


Solution:Substitute t= 20 into the formula to find the value for P.

P =100 e0.075(20)

= 100 e1.5= 100 (0.22313)= 22.313

6.5.2 Investment with Compound Interest

The total amount of money, denoted by Sis the compound amount for a sum ofmoney Pcompounding after n-th year, where the interest is payable k times atthe rate ofr% per annum, is given by the formula below:

1

nkr

S Pk

= +

Where:S = compound amount or the prospective valueP = initial investment or the principal valuer = interest rate per annumk = number of interest paid (compound) in a year

n = number of year

Example:If RM1000 is invested at the rate of 6% per annum, compounding (payable) everyquarterly, what would the total amount be in the account after 10 years?

Solution:S =?, P= 1000, r= 6% = 0.06, k= every quarterly = 4 a year, n= 10

Then

( )( )

( )

( )

10 4

40

1

0.061000 1

4

1000 1.015

1000 1.81402

1814.02

nkr

S Pk

S

S

S

S

= +

= +

=

=

=


16/18


Example:Determine the principal amount of a loan, given that the prospective amountpayable after 10 years is RM21,589.20 and the compound rate of 8% per annum,

compounding (payable) on yearly basis.

Solution:S =21,589.20, P= ?, r= 8% = 0.08, k= every yearly = 1 a year, n= 10

Then

1

nkr

S Pk

= +

( )( )

( )

( )

10 1

10

0.0821589.20 1

1

21589.20 1.08

21589.20 2.15892

21589.20

2.15892

10000

P

P

P

P

P

= +

=

=

=

=


17/18


1. (a) Given that the price of one acre of land is increasing at therate of 2% per year. How long will it take for the price toincrease to RM30,000, given its current value is RM10,000.

(b) Due to the economy downfall, the total number of residents in atownship is reducing at the rate of 1% per year. Initial populationwas 100,000 residents. What is the population after 3 year?

2. Determine the compound amount, given the following principalvalues, compound interest rates and time periods:

(a) RM5500; 6% per annum compounding on monthly basis; 18months.

(b) RM10,000; 8% per annum compounding yearly; 5 years.(c) RM7600; 7.26% per annum compounding on quarterly basis;

5 years and 8 months.

(d) RM2300; 5.75% per annum compounding daily; 150 days.(assume 1 year = 365 days)

3. Determine the principal amount, given the following compoundvalues, compound interest rates and time periods:

(a) RM16,084.82; 6% per annum compounding monthly; 14months.

(b) RM10,197.02; 5.3% per annum compounding daily; 135 days.(assume 1 year = 365 days)

(c) RM6,657.02; 12.6% per annum compounding every 2 months;10 months.

(d) RM36,361.63; 7.2% per annum compounding every 3 months;5 years and 3 months.

EXERCISE 6.4


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Exponential function and its inverse, i.e. logarithm function, form graphsreflection upon a liney = x.

The properties of exponentials and logarithms have to be grasped whentackling problem solving questions.

In addition, the skills to convert equations in the form of exponential tologarithm form, and vice-versa are equally significant.

Exponential function Logarithm function

Visit the following websites:

(i) http://webmath.amherst.edu/qcenter/logarithms/index.htmlfor questions regarding logarithm;

(ii) http://www.webster-on-line-dictionary.org/definitions/english/LO%5Clogarithm.html for further discussion; and

(iii) http://www.okc.cc.uk.us/maustin/Log-Functions/Logarithm%20Functions.htm for samples on solving logarithm problems.

ACTIVITY 6.5

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