# WORKING WITH LOGARITHMS

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```					WORKING WITH
LOGARITHMS
MTA Cole Harbour High School 2009

10/23/2009
David Mac Farlane
Dr. J. H. Gillis Regional High School
WORKING WITH LOGARITHMS

Inverse of a Function

Graph                                             6   y                   Graph                                                6   y
y  2x  1                                                                x  2y 1
x         y                                    4                          y        x                                        4

2                                                                            2
x                                                                            x

-6    -4       -2               2       4   6                            -6    -4           -2               2       4   6
-2                                                                           -2

-4                                                                           -4

-6                                                                           -6

What have you noticed concerning the x and y values __________________________________
2                                                                        2
Graph y  x                                       6   y                   Graph x  y                                          6   y

x         y                                    4                          y        x                                        4

2                                                                            2
x                                                                            x

-6    -4       -2               2       4   6                            -6    -4           -2               2       4   6
-2                                                                           -2

-4                                                                           -4

-6                                                                           -6

What have you noticed concerning the x and y values __________________________________
Solve the following for y, x  2 y  2                                                                        2
Solve the following y, x  y

Graph the function
Graph the function
6    y
6        y
4
4
2
2
x
x
-6        -4    -2                 2       4       6
-6        -4    -2                     2       4       6
-2
-2
-4
-4
-6
-6

What have you noticed ______________________________________________________________

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WORKING WITH LOGARITHMS

Graph                                                                    Graph
x                                                                         y
y  10                            1000       y                            x  10        3    y
x       y                                                                 y       x
800                                                  2

600
1

400                                                                                   x

200   400   600   800   1000
200
x                      -1

-4   -3   -2    -1            1       2   3   4                      -2

What have you noticed concerning the x and y values

y
Solve x  10 for y:

Is there a problem

Graph the equations y  2 x and x  2 y
30       y                                     y=x
28
26
24
22
20
18
16
14
12
10
8
6
4
2                                                             x

-4 -2
-2             2   4   6   8 10 12 14 16 18 20 22 24 26 28 30
-4

Fold this graph along the y  x line what do you notice

Find the inverse for the following equations

1
a) y            x2                b) y  3 x  2                       c) y  2 x2  4
4

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WORKING WITH LOGARITHMS

The logarithm was perhaps the single most useful arithmetic concept in all the sciences:
and an understanding of them is essential to an understanding of many scientific ideas.
Logarithms may be defined and introduced in several different ways. But for our purposes,
let‟s adopt a simple approach. This approach originally arose out of a desire to simplify
multiplication and division to the level of addition and subtraction. Of course, in this era of
cheap hand held calculators, this not necessary anymore but it still serves as a useful way
to introduce logarithms. The question is, therefore:

Is there any operation in mathematics which produces a multiplication by the
performance of an addition? The answer is simple, exponents as shown with the
simplification of

*Note: the addition will only work if the bases are the same in the case of        we cannot

In general, the addition rule can be written as                as was seen in the previous
section on exponents. This expression will do our job of multiplying any two numbers such
as 1.3 and 6.9, if we can only express 1.3 as   and 6.9 as .

What number shall we use for the base? Any number will do, but traditionally, only two are
in common use. Ten (10) an the transcendental number e (=2.71828…), giving logarithms
to the base 10 or common logarithms (log), and logarithms to the base e or natural
logarithms (ln). We will deal exclusively with logs and will leave the natural logarithm for
future study.

Lets talk about logarithms to the base 10 or the common logarithm. We thus choose to let
our number 1.3 be equal to

„a‟ is called “the logarithm of 1.3”. How large is „a‟? Well, its not 0 since   and it‟s less
than 1 since            . Therefore, we see that all numbers between 1 and 10 have
logarithms between 0 and 1. If you look at the table below you‟ll see a summary of this.

Number Range                Logarithm Range
0–1
1–2
2–3
etc                            etc
You see, we have the number range listed on the left and the logarithm range listed on the
right. For number between 1 and 10, that is between 100 and 101, the logarithm lies in the
range 0 to 1. For numbers between 10 and 100, that is between 101 and 102, the logarithm

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WORKING WITH LOGARITHMS

lies in the range 1 and 2, and so on. Log tables use to be used to solve these, now we just
use the calculators.

Most calculators are very straight forward in obtaining the logarithm. They either have
two logarithm key or a dual function key. In any case, the labels will be „log‟ and „ln‟ which
is often pronounced „lawn.‟

Log is the key or logs to the base 10 and ln is for natural logs. We want logs to the base 10
in our example so we use „log‟.

Log(1.3)=0.11394…         which means 1.3 =

Log(6.9)=0.83884….        which means 6.9 =

Therefore,

1.3  6.9 
10.11394 10.83884  10.11394.83884 
100.95278

100.95278  8.969 and 1.3  6.9 = 8.97

The problem of finding a number when you know its logarithm is called finding the
“antilogarithm or sometimes “exponentiation”. Again, let‟s look at your calculator rather
0.95278
than having to place 10       by entering 10^(0.95278). We can take the inverse of the log
or the antilogarithm of 0.95278. You will find that it will result in the answer 8.97 as well.

The logarithmic and exponential function are very important since many physical and
biological processes can be described by them.

Complete the following simple calculation using logarithms

18  7       log18  ___ ;log 7  ___                   ___  ___  ____      10 y  ____
a                 b                 a     b       y
189 12      log189  ___ ;log12  ___                  ___  ___  ____      10 y  ____
a                 b             a     b       y

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WORKING WITH LOGARITHMS

Definition of a logarithm
We noticed belfore that we had difficulty in trying to solve the equation x  10 for y.
y

How can we get the y from the exponent into the form y  ? This is where the logarithm
proves it usefulness.
The logarithm is basically the exponent of a particular base, for example the exponent of
100 when expressed with a base of 10 is 2 or 100  10 . We would say that the log of 100
2

base 10 is 2. This is written in the form
log10 100  2 where we know 102  100
When the base is 10 it is referred to as the common logarithm as it is the one used most
frequently. Due to this the base is omitted to simplify calculations and would typically
appear as;
log100  2
Other examples of logarithms would be;

   The log of 16 base 4 is 2, in other words       4x  16, x  2 or log416  2
   The log of 16 base 2 is 4, in other words          2x  16, x  4 or log216  4

DEFINITION OF LOGARITHMS

The general rule for common logarithms:
log a  c  a  10c
The general rule for logs base b:
logb a  c  a  bc
Restrictions logb a  c, a  0, b  0, b  1

We should also note at this time that an exponent and logarithm of the same base are
actually inverses of each other, as they are reflected across the line y  x .
10     y
y=x

y  bx
5

y  log bxx

-10      -5                     5            10

-5

-10

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WORKING WITH LOGARITHMS

Graphing exponential equations in the form

Complete the tables below and graph the result on the grids provided for the equation

6         y                                                            6    y

4                                                                      4
-2                                                                         -2
-1                  2
x
-1              2
x
0                                                                          0
5               10                                    5       10
1                                                                          1
-2                                                                     -2
2                                                                          2
3                   -4                                                     3               -4

-6                                                                     -6

Asymptote equation _______________________                                  Asymptote equation ________________
6   y                                                    6    y

4                                                        4
-2                                                                         -2
-1                                2                                        -1              2
x                                            x
0                                                                          0
1              -4        -2               2           4   6           8
1                            5       10
-2                                                           -2
2                                                                          2
3                             -4                                           3               -4

-6                                                           -6

Asymptote equation _________________                                        Asymptote equation ________________
4     y                                                                     y
2
2                                                                                           x
-3                                                            x                                         5       10
-2                                            5               10           -2              -2
-1                  -2                                                     -1
0                                                                          0
-4

1                   -4
1               -6
2                   -6                                                     2               -8
3
-8

Asymptote equation ________________                                         Asymptote equation ________________

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WORKING WITH LOGARITHMS

Must know exponent rules

bx
 
y
1. b  1
0
2. b  b  b
x    y        x y           3.
y
 b x y        4. b x           b xy
b
5.                                          x              x
a             b                7. log x x  1                8. log b 1  0
b   x

1       6.                                                                                               9. logb ba  a
bx             b             a

Log rules you must know                                                                                                                Helpful log rules
1
1. log ab  log a  log b                                                                                                              1. logb a 
log a b
a                                                                                                                      2. logb a   log 1 a
2. log             log a  log b
b                                                                                                                                                   b

3. log a  n log a
n
3. b   log b a
a
.
1
log b a 
log k a
, where k is the desired base                                                                                        4. log b           log b   x
x
log k b
#4 is extremely useful and is called the                                                                                                    log a x logb x
5.          
CHANGE OF BASE FORMULA                                                                                                                      log a y logb y

Graphs you should know

5    y
y  ax , a  1                                      5    y
y  a x ,0  a  1
4                                                                                       4
3                                                                                       3
2                                                                                       2
1
x                                                           1
x
-2                     -1                         1              2
-2          -1                         1             2
-1
-1
2        y                                                                                       3   y                                            y  log a x, 0  a  1

1                                                          y  log a x, a  1                    2
x
1
-1                               1        2            3      4                                                                                          x
-1
-1                   1            2        3         4
-2                                                                                               -1

-3                                                                                               -2

-4                                                                                               -3

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WORKING WITH LOGARITHMS

LIST OF WRONGS AND RIGHTS

The most important rule is not to make your own rules!

WRONG                                            RIGHT
1      log a    n
 n log a                1     log a    n
cannot be simplified

2     log a  log b  log ab                   2    log a  log b cannot be simplified

3     log(a  b)  log a  log b               3    log(a  b) cannot be simplified

4     log x  log x  log x2                   4    log x  log x   log x 
2

5     log y x  log x  log y                  5                    log b x
log y x 
log b y
6     log x                                    6    log x
= log x  log y                               = log y x
log y                                         log y
7           a log a                            7          a
log                                          log      log a  log 2
2   2                                         2
8     log10 10       log10                     8    log10
=  2 or        =log2                          =log510  1.432
log 5 5        log 5                          log 5
9     log 2 5                                  9    log 2 5
 log 2                                       cannot be simplified
5                                             5
10    b x  b y  b x y                       10   b x  b y cannot be simplified

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WORKING WITH LOGARITHMS

Logarithms worksheet #1

L1. log 2  log 3  log N                             L23. log N  log5  1
L2. log 3 5  log 3 6  log 3 N                       L24. log 3 N  log 3 4  2
L3. log 7 15  log 7 3  log 7 N                      L25. log 5 x3  log 5 x  2
L4. log N  log 4  log 20                            L26. log 9 x  log x  4
L27. log 6 N  log 6 ( N  1)  log 6 3
L5. log 1 N  log 1 2  log 1 3  log 1 1             L28. log 7 ( x  1)  log 7 ( x  5)  1
2          2        2         2
L29. log 6 ( y  3)  log 6 ( y  2)  1
L6. log       N  log 2 3  log 2 6  log         2
L30. log 2  9t  5  log 2  t 2  1  2
2                                   2

L7. log8 m  log8 m  l og8 4
L8. log N  log3  log 4  log 6                      L31. log3  2v  5  log3  v2  4v  4  2
L9. log  x2  2 x  15  log  x  3  log8        L32. log( N  5)2  log( N  5)  log 2
L10. log 2 14  log 2 3  log 2 6  log 2 x           L33. log N  2log N  log8
L11. log10  x2  1  1
2
L34. log5 3 N  log5 N  3
3
2
L12. log x    1                                     L35. 4log 3 N  log 3 N 2  log
1
3                                                                           8
1
L13. log 1  x                                             1          2
L36. log N  log N  3log N  log16
4
2                                              3          3
L14. log N  2log5                                    L37. log 3 81  x
L15. log 5 N  3log 5 2                               L38. log16 0.125  x
1                                                    x
L16. log 7 N  log 7 36                                    1
2                                     L39.    27 x 1
1                                             81 
L17. log N  log8                                     L40. log2  log16 log3 x  1
3
L18. 3log N  log125                                         solve for x
L41.
L19. 2 log 8 N  log 8 49                                    log 6 ( x  4)  2  log 6 x
1                                    L42. Solve for x
L20. log x 2 
6                                    log 2 ( x  2)  log 2 x  log 2 ( x 2  6)
L21. log 5 25  x                                     L43. Solve for x
L22. log10 ( x  1) 2  2                            log 2 (2  2 x)  log 2 (1  x)  5
L44 For the graph of the function
y  4  2 x 1 , give the a)domain
b) x-intercept c) equation of
asymptote

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Applications of Exponents and Logarithmic Functions

Compound Interest

1. How much should you invest at 7% interest per annum compounded annyally so
that \$5000 will be available in 4 years?
2. In 1626 Manhattan Island was sold for \$24. If that money had been invested at 8%
per annum compounded annually what would it be worth today?
3. Suppose you invested \$200 at 9% per annum compounded annually. How many
years would it take for your investment to grow to \$500?
4. Mary invests \$2500 at 11%per annum compounded monthly. How many years will
it take her investment to double in value?
5. Laurel invests \$1200 at 8% interest compounded annually. Marco invests \$900 at
11% interest compounded annually. To 2 decimal places how long will it take before
Marco has as much money as Laurel

Light penetration

6. For every metre a diver descend under water, the intensity of the three colors of
light is reduced as shown
a. For each color, write the equation which           Color Percent
expresses the percent P of surface light as a              Reduction
function of the depth d metres.                           Per metre
Red     35%
b. For each color, determine the depth at which
Green 5%
about half the light has disappeared.             Blue    2.5%
c. Let us agree that, for all practical purposes,
light has disappeared when the intensity is only 1% of that of the surface at
what depth would this occur for each color
7. Several layers of glass are stacked together. Each layer reduces the light passing
through it by 5%.
a. What percent of light passes through 10 layers of glass?
b. How many layers of glass are needed to reduce the intensity of light to 1% the
original.

Population Growth

8. The town of Springfield is growing at a rate of 6.5% per annum. How many people
are there in Springfield now, if there will be 15 000 in 4.5 years?
9. A culture has 750 bacteria. The number of bacteria doubles every 5h. How many
bacteria are in the culture after 12 h?
10. A colony of bees increases by 25% every 3 months. How many bees should Raimain
start with if he wishes to have 10 000 bees in 18 months?
11. If the population of a bacteria culture doubles every 30 minutes how long will it take
to triple?
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12. Prove that if the population growth rate is constant, the time required for a
population to double is independent of the population size.

Nuclear Fallout

13. When strontium-90 decays, the percent remaining is expressed as a function of the
time t years by the equation                   . How long is it yntil the persend
remaining is
a. 10% b. 1%
14. The halflives of two products of a nuclear explosion are
shown. For each substance                                       Substance Halflife
a. Draw a graph showing the percent remaining                Iodine        8.1 days
during the first five halflives.                         Cesium        282 days
b. Express the percent remaining as a function of:
i. The number of halflives elapsed, n
ii. The number of days elapsed, t.
c. What percent of the substance remains after:
i. One week
ii. 30 days
iii. One year?
d. How long until the percent remaining of each substance is:
i. 10%
ii. 1%

Other

15. Tom bought a new car for 15 000 dollars. Each year the value of the car depreciates
by 30%. In how many years will the car be worth only \$500 dollars?
16. A ball bounces to 85% of its original height with each bounce. If we agree that a ball
stops bounces when it reaches 0.1% of its original drop height. How many bounces
will it take for the ball to stop bouncing?
17. A barely audible sound has a decibel level of 0 and is denoted by Io. If I equals the
intensity of a sound then the decibel level is
1
a. If one stereo is playing at 75 decibels and a second stereo is playing at the
same time also at 75 decibels, what is the decibel level of the two stereos
playing simultaneously?
b. What would the decibel level be if there were three stereos simultaneously
playing at 75 decibels?

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Logarithm Worksheet #2
As easy as falling of a Log
log12log 4                                                                      1
1. 3 81  27                     2. 10                             3. 5
log 8 2
4. log 3  
9
1        1                 6. log 5 0.008                         2         4                              3
ab5
5. 27 4      98                                                   7. 2 3     16 3                       8.
4
a 3b
3                                 2                                  2                                     3
9. 4 2                           10. 27 3                          11. 32 5                              12. 25 2
3
14.   234                        15.      3
5  25                       16.         535
13. 16 4
log8log 2                      log 2                                          1
17. 10                           18. 3 4                           19. log 2                            20.         ab3  a2b
3

8
2        1                       1        1                       5           5                    24. 4 32  3 2
21. 2 3      46                             1 4                 23. 2 6       32 6
22. 27 2   
9
25. log 2 16 8                            1 
26. log x    5                 27. log x 81 
4
28. log 4 x  
3
 32                                           3                               2
29. log    x0                   30. 2
x 2
 64                                      1                     log 9  a, then
10                                                         31. log x                            32.
 16                    log 3  ___
33.                                         34.                                                35. log( x2 1)  log( x 1)  1
log 2 32  log 4 8  log 2 x  0                   2        1
log N  log 64  log16
3        2
36. log 4 3  log 4 ( x  2)  2            37.                                                      log( x  6)
38.               2
1                                       log x
log5 N  2log5 27  log5 9
3
2
40. 4
2log 4 x
7                                  41. log(log x)  2
39. 35 x  9x  27
x 1                                                                        1 x
43. log a a x  x4  2                                       92 x
2

1                                                                                  44. 3
42. 4   
2x
8
45. log5 0.04  x                                        1
x 1
47. 6
3log 6 x
8
46. log8 x   
8
48.            4n 10  16(4n )  0              1
x
 1 
2 x 1
49.     25 
x2
 125                  50. 93 x     
5                                                          27 
x 1
51. 4
3 2 x
 16        3x
52. 3
x 2
3 5
53. 4     23 x 1
54. log 2 (log 2 x)  2                     55. 2
x2 2 x
8                                   56. log100 1  x

57. log 
 1 
  2                             
58. log 3 x 2  7  2                             59.
log6  n  4  log6  n  9  2
 3n  4 

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60. logb b x 2 x5  10
2
1           62
        
61. log 6 n 2  64 2  1                  3       1
log x  log81  log 36  log 2
4       2
63. log(n  4)  log n  log5     64.                                65.
log x  log 5  3log 3  2 log 9   log3  x  3  log3 (2x  8)  2
66. log(3n)  log n  log 2  0   67.
log( x2  9)  log( x  3)  1
Write the following in logarithmic form
mv 2                              V                                  s2 3
68. F                            69. r                             70. A 
r                                h                                   4

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MATHEMATICS 12 LOG ASSIGNMENT ANSWER KEY
NAME__________________
Must hand in your working paper showing the work you have completed along with this key.

1.               2.               3.               4.              5.

6.               7.               8.               9.              10.

11.              12.              13.              14.             15.

16.              17.              18.              19.             20.

21.              22.              23.              24.             25.

26.              27.              28.              29.             30.

31.              32.              33.              34.             35.

36.              37.              38.              39.             40.

41.              42.              43.              44.             45.

46.              47.              48.              49.             50.

51.              52.              53.              54.             55.

56.              57.              58.              59.             60.

61.              62.              63.              64.             65.

66. _______

67. _______

68. ___________________________________

69. ___________________________________

70. ___________________________________

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TEAM NAME _________________________________________________
(Must have some mathematical relevance i.e. The Derivers)
Team members ______________,______________,________________,_______________
Envelope _____         Envelope _____          Envelope _____

L              ____________________
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Envelope _____         Envelope _____         Envelope _____

O              ____________________
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16 | P a g e
WORKING WITH LOGARITHMS

LIST OF WRONGS AND RIGHTS

The most important rule is not to make your own rules!

WRONG                                    RIGHT

1       log a n  n log a                  1

2      log a  log b  log ab                2

3      log(a  b)  log a  log b            3

4      log x  log x  log x2                4

5      log y x  log x  log y               5

6      log x                                 6
= log x  log y
log y
7            a log a                         7
log     
2   2
8      log10 10       log10                  8
=  2 or        =log2
log 5 5        log 5
9      log 2 5                               9
 log 2
5
10     b x  b y  b x y                    10

17 | P a g e
WORKING WITH LOGARITHMS

Log worksheet 1 solutions
L1         6           L13   0.5   L25    5      L37    4
L2         30          L14   25    L26           L38

L3             5      L15    8     L27           L39    -3

L4                    L16    6     L28    6      L40    81

L5             6      L17    2     L29    4      L41

L6             9      L18    5     L30    3      L42    3
L7             2      L19    7     L31    7      L43    -3
L8             8      L20    8     L32    -3     L44a
L9                   L21    4     L33    2      L44b   3
L10            28     L22          L34    125    L44c   -4

L11            3      L23    2     L35

L12            1.5    L24    36    L36

18 | P a g e
WORKING WITH LOGARITHMS

MATHEMATICS 12 LOG ASSIGNMENT ANSWER KEY
NAME__________________
Must hand in your working paper showing the work you have completed along with this key.

6 17                           3
1.    3     2.     3         3.         5      4.     -2       5.    3
b 12 5 7
8.         b a
6.         -3     7.     64               a          9.     8        10.   9
1
12.    
11.        -4              125     13.               14.             15.

16.               17.              18.               19.             20.

21.               22.              23.               24.             25.

26.               27.              28.               29.             30.

31.               32.              33.               34.             35.

36.               37.              38.               39.             40.

41.               42.              43.               44.             45.

46.               47.              48.               49.             50.

51.               52.              53.               54.             55.

56.               57.              58.               59.             60.

61.               62.              63.               64.             65.

66. _______

67. _______

68. ___________________________________

69. ___________________________________

___________________________________

19 | P a g e

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