# Logs - ROHAN Academic Computing - SDSU by cuiliqing

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```									      Logarithms
This series of slides demonstrates the usage of logarithms. Logs have a
variety of applications in economics and finance.

To obtain the spreadsheet that contains the examples click on the following:
ftp://rohan.sdsu.edu/faculty/vandenberg/Logs.xls
ms. Logs have a

lick on the following:
How do logarithms work?

ROW/COL              B                    C                    D                  E
4
5
6                              Number 1               Number 2         Result of Multiplying
7       Number                       10                    10                  100
8       Power of the number           1                     1                    2
9
10      Notice we get the same result by multiplying the numbers as we do by adding the power of the numbers.
11      The power of the number is also its exponent. By default we do not write the power if it is 1.
12
13      Works for any numbers                                          Result of Multiplying
14      Number                         100                  10                 1000
15      Power of the number             2                    1                   3
16
17      We also that we can write:                       1,000,000
18      as                                                 #NAME?
19      or                                                 =10^6       (remember an ^ means raised to the power)
20
21      Thus
22      Number                       1,000,000           1,000,000       1,000,000,000,000
23      Power                            6                   6                   12
24
25      In fact the spread sheet might use:         1E+12        to represent the number.
26      As you can see the E+12 means move the decimal point 12 space to the right and you get the answer.
27
28      It would not be hard to figure the exponent for any number which is an even power of 10
29
30      So then we can see that                           100,000
31      is the same as                                       #NAME?
32      or                                                 =10^5
33
34      Rather than count the number of 10's need to get the number a spreadsheet provides a function to do this
35
36
Log of number
(or the number
37                             Number                 of 10s)          Formula
38                                               10            1             #NAME?
39                                              100            2             #NAME?
40                                             1000            3             #NAME?
41                                            10000            4             #NAME?
42                                           100000            5             #NAME?
43                                          1000000            6             #NAME?
44
45      The log function is particularly helpful if we wish to find the power of 10 when the number is not exactly a
46      We can write                   12,345
47      as                              =10^4.09149109426795
48      or the log of                  12,345        = 4.09149109426795
49
50   Works for any numbers
51                              Number 1          Number 2       Result of Multiplying
52   Number                       12,345             678              8,369,910
53   Power of the number       4.091491094       2.831229694        6.922720788
54
55
56   In days before electronic calculators and computers logs were commonly used because addition is easier
57   Tables of logarithms were available as were slide rules which were base on logs.
58
59   To reverse the process, to convert a logarithm into a number, you need to do the opposite.
60   Again for even powers of 10 that is not difficult. For uneven powers of 10 use the spreadsheet power symb
61
62   Log of a number is:       4.091491094
63   Exponent form:               =10 ^ 4.09149109426795
64   The number is                12,345          #NAME?
65
66   Therefore             10^(log(Number)) = Number
67   Example               Number
68                               12,345          12,345                #NAME?
F               G         H   I

Mathematically we can say:
=10*10
=1+1

adding the power of the numbers.
te the power if it is 1.

Mathematically we can say:
=100*10
=2+1

an ^ means raised to the power)

=1000000*1000000
=6+6

nt the number.
e right and you get the answer.

even power of 10

sheet provides a function to do this

0 when the number is not exactly a power of 10
Mathematically we can say:
=12345*678
=4.09149109426795+2.83122969386706

nly used because addition is easier than multiplication

to do the opposite.
10 use the spreadsheet power symbol (^) to do it.
Log Scales
ROW/COL                      B                     C                   D             E        F      G
4            Frequently data is hard to understand without using logs
5            Here is some Forecasted data for the next few years. Look at the graphs below
6
7
Alternative
log of       way to
8                        Year                Forecast           number     express the # #NAME?
9                            2000                          10       1          10 ^1
10                            2001                         100       2          10 ^2
11                            2002                       1,000       3          10 ^3
12                            2003                      10,000       4          10 ^4
13                            2004                     100,000       5          10 ^5
14                            2005                   1,000,000       6          10 ^6
15                            2006                  10,000,000       7          10 ^7
16                            2007                 100,000,000       8          10 ^8
17                            2008               1,000,000,000       9          10 ^9
18                            2009              10,000,000,000      10         10 ^10
19                            2010             100,000,000,000      11         10 ^11
20                            2011           1,000,000,000,000      12         10 ^12
21
22
23
24                                   Here is one picture of the data
25                                   Notice that it is hard to see what is going on.
26
27
28                                                       Original Data
29
30
31                         1,200,000
Millions

32
33                         1,000,000
34
35
36                             800,000
37
38                             600,000
Values

39
40
41
400,000
42
43                             200,000
44
45                                    0
46
47                                     1995              2000              2005          2010          2015
48                             -200,000
49
50
51
Time
52
53
54
55                         Here is a picture using the log scale
56                         Notice that you can see the series is going straight up.
57
58
59
60
61                                       Using the log scale of the Y axis
62
63
64
65
66                         1,000,000,000,000                                                    10 ^12
67                           100,000,000,000                                                  10 ^11
68
10,000,000,000                                               10 ^10
log values are graphed

69
70                             1,000,000,000                                             10 ^9
71                               100,000,000                                          10 ^8
72                                10,000,000                                        10 ^7
73
74
1,000,000                                     10 ^6
75                                   100,000                                   10 ^5
76                                    10,000                                10 ^4
77                                     1,000                              10 ^3
78
79
100                           10 ^2
80                                        10                         10 ^1
81                                         1
82
83
1995                2000           2005      2010        2015
84                                                                             Time
85
86
87
88                         Using the logs of the data rather than a log axis
89                         You see the same effects
90
91
92
93
Transformed data
94
95
96                         14
97
98                         12                                                                           10 ^12
the log of the values

99                                                                                                 10 ^11
100
10                                                                 10 ^10
101
10 ^9
Using the log of the values
102                                                                                10 ^9
103                             8                                             10 ^8
104                                                                      10 ^7
105                             6                                   10 ^6
106
107
10 ^5
108                             4                         10 ^4
109                                                  10 ^3
110                             2               10 ^2
111                                        10 ^1
112
113
0
114                              1998   2000    2002      2004          2006   2008        2010   2012
115                                                              Time
H   I

2015
This result was
achieved by right
clicking the axis and
then choosing to
check this log box.

10 ^12
2012
The famous log scale.
If you live in California you should be familiar with a famous log scale.
It is called the Richter Scale, it measures an earthquake's power.

So an earthquake with a Richter reading of    4
has a certain power. If another earthquake with a Richter reading of        5
occurs, how much more powerful is it.

Since Richter readings are in log form then a 5 is 10 times more powerful than a 4

So the difference between a 3 and a 7 magnitude is substantial.
A 7 is 10,000 more powerful than a 3.

http://neic.usgs.gov/neis/general/handouts/richter.html
owerful than a 4
Other bases for logs.
ROW/COL         B                 C            D           E         F   G          H         I
4
5
6      There is nothing magic about base 10 (or common logarithms). You could build logs on any base you want.
7      In fact the log function has an option to set the base to any you may want to use.
8
9      Number             log to the base 2 of the number
10                     2         1      =2 ^ 1
11                     4         2      =2 ^ 2
12                     8         3      =2 ^ 3
13                    16         4      =2 ^ 4
14                    32         5      =2 ^ 5
15                    64         6      =2 ^ 6
=log(number, base)
16
17      Other bases
18                                         Base
19         Number                    2       5        10        16
20                    2         1.0000 0.4307     0.3010    0.2500 #NAME?
21                    4         2.0000 0.8614     0.6021    0.5000
22                    5         2.3219 1.0000     0.6990    0.5805
23                   10         3.3219 1.4307     1.0000    0.8305
24                   16         4.0000 1.7227     1.2041    1.0000
25                   20         4.3219 1.8614     1.3010    1.0805
26                                              #NAME?
27      The blue numbers are the logs of the number for different bases
28      Notice that when the base is equal to the number the log is always 1
29
30      Natural logarithms
31
32      Look at the following expansion:

33            X             (1+1/X)^X
34                   1         2.00000      As X expands the
35                   2         2.25000    value of expression
36                   3         2.37037     increases, but at a
37                   4         2.44141     slower and slower
38                   5         2.48832     rate and reaches a
39                   6         2.52163    limit. This limit is a
40                   7         2.54650
number which like
41                   8         2.56578
pi, never rounds off
42                   9         2.58117
even. This number
43                  10         2.59374
is called "e." It has a
44                1010         2.71694
lot of useful
45                2010         2.71761
46                3010         2.71783        mathematical
47                4010         2.71794   properties. Because
48                5010         2.71801       of this it is used
49                6010         2.71806      frequently. As an
50                7010         2.71809         example the
51                8010         2.71811    derivative of e^x is
e^x. Very
convenient.
mathematical
properties. Because
of this it is used
frequently. As an
example the
derivative of e^x is
52             9010      2.71813         e^x. Very
53            10010      2.71815       convenient.
54            11010      2.71816
55            12010      2.71817
56     very large 2.7182818284591                       <==Value of "e" used by Excel
57        1000                                           #NAME?
58   Natural logarithms use "e" as their base.
59   So the log to the base e is the value you need to raise e to get the number
60
61   Spreadsheets use a special set of functions to manipulate the natural logs.
62
63   To get the natural logarithm use =ln(number)
64   To get the anti-logarithm (to reverse the process) use =exp(mumber)
65
66
67                  Number        ln(number)                                         Result
68                            1    0.000000 = 2.71828182845905 ^ 0 =                         1
69                            2    0.693147 = 2.71828182845905 ^ 0.693147180559945 =         2
70                            5    1.609438 = 2.71828182845905 ^ 1.6094379124341 =           5
71                            7    1.945910 = 2.71828182845905 ^ 1.94591014905531 =          7
72                           10    2.302585 = 2.71828182845905 ^ 2.30258509299405 =         10
73                           25    3.218876 = 2.71828182845905 ^ 3.2188758248682 =          25
74
75   So again exp(ln(number)=number
76
77                  Number
78                       1000         1000 #NAME?
79
80
81
82
83
gs on any base you want.
Natural logs are frequently used in finance.
ROW/COL            B               C              D            E             F          G          H
5
6
7      The compound value of a \$ is:
8
9
10
11
12                                   V = P *(1+ i/n)^(n*t)
13
14      where:
15
16      V is the ending value
18      I is the rate of interest
19      n is the frequency of compounding
20      t is the length of time
21
22      Notice that part of the formula looks a lot like the expansion to get the value of "e"
23      If we let P=1 and t=1 it is exactly the same formula.
24
25
26
27
28                                   V = 1 *(1+ i/n)^(n*1)
29
30      If n =1 then we get: (which is frequently the form of the compound amount formula)
31
32
33
34
35                                   V = P *(1+ i/1)^(1*t)
36      Impact of compounding
37      Using an interest rate of:                       5.0000%
38      Assume the time period is 1 year and the amount is \$1
39
40      Frequency of compounding                n           V       Annual Rate of return
41      Annual                                     1        \$1.05000 5.0000%
42      Semi-Annual                                2        \$1.05063 5.0625%
43      Quarterly                                  4        \$1.05095 5.0945%
44      Monthly                                   12        \$1.05116 5.1162%
45      Daily                                  365          \$1.05127 5.1267%
46      Hourly                                8760          \$1.05127 5.1271%
47      Minutely                            525600          \$1.05127 5.1271%
48      Secondly                          31536000          \$1.05127 5.1271%
49      Infinite                          =EXP(i)           \$1.05127 5.1271%
50
51      Notice that as the frequency of compounding increase our rate of return increase, but at a decreasing rate.
52   Until the difference between daily, secondly and infinite is nil.
53
54
55   So if you allow infinite compounding the future value of a dollar formula becomes:
56
57
58
59
60                                        V = P*exp(i*t)
61
62   Remembering that ln and e^x are opposite transactions we can determine an equivalence between rates of i
63
64
65   You would indifferent between the following rates compounded once a year, or the rates compounded conti

Compounded     Compounded          Compound amount of \$1
66   once a year    continuously          at the end of a year
ln(1+Annual
Annual Rate of     Rate of          Once a
67      return          Return)           year      Continuously
68     5.0000%         4.8790%           \$1.0500      \$1.0500
69     5.0625%         4.9385%           \$1.0506      \$1.0506
70     5.0945%         4.9690%           \$1.0509      \$1.0509
71     5.1162%         4.9896%           \$1.0512      \$1.0512
72     5.1267%         4.9997%           \$1.0513      \$1.0513
73     5.1271%         5.0000%           \$1.0513      \$1.0513

74
75   You can always find a rate that will cause you to be indifferent between an annual rate and continuously com
76
Continuously
77   Annual Rate    Compounded
78          5.0000%       4.8790% #NAME?
79
80   Or you find a continuously compounded rate that is equal to an annual rate by using exp(i)-1
81
Continuously
82   Compounded     Annual Rate
83          5.5000%      5.6541% #NAME?
84
85   Try it by changing the green numbers
86
87
88   In many cases theoretical consideration require that we use continuously compounded rates in finance.
89   In general when using the CAPM and Black/Scholes you should use continuously compounded rates.
90
I

ln(Annual Rate of Return)
4.8790%
4.9385%
4.9690%
4.9896%
4.9997%
5.0000%
5.0000%
5.0000%
5.0000%

e, but at a decreasing rate.
uivalence between rates of interest and compounding

the rates compounded continuously.

al rate and continuously compounded one by taking the ln(1+i)

sing exp(i)-1

ounded rates in finance.
ly compounded rates.
The fundamental equation of finance:
ROW/COL    B               C                 D           E      F     G   H   I
5
6
7           The compound amount equation is:
8
9
10
11                               V = P *(1+ i)^t
12
13      This equation can also be stated as follows.
14
15
16
17                    ln(V )= ln(P)+t*ln(1+ i)
18
19
20           I=                          10.00%
21           P=                          \$100.00
22           t=                               10
23
24           ln(v)=                     5.558272 #NAME?
25
26           V=                          \$259.37 #NAME?
27
28           Using the FV function       \$259.37    #NAME?
29
30           An alternative would be:
31
32
33
34
35
36                           ln(\$1.00 )=t*ln(1+ i)
37
38
39           I=                           10.00%
40           P=                             \$1.00
41           t=                                10
42
43           ln(1)=                     0.9531018 #NAME?
44           FV Factor                  2.5937425 #NAME?
45
46           For                           \$100
47           V=                          \$259.37 #NAME?
48
49
50           Another alternative for annual compounding would be:
51
52
53
54
55
56                V=P*exp(t*ln(1+ i))
57
58   I=                        10.00%
59   P=                        \$100.00
60   t=                             10
61
62   V=                        \$259.37 #NAME?
63
64   For continuous compounding the formula would be:
65
66
67
68
69                        V=P*exp(t*i)
70
71   I (continuously) =        10.00%
72   P=                        \$100.00
73   t=                             10
74
75   V=                        \$271.83 #NAME?
76
77
78   Note the difference of       \$12.45 #NAME?
79   is caused by the frequency of compounding
80
81
82
83
84
Stock Returns
ROW/COL       B          C           D                 E                    F             G
3
4       As you know many things in this world are normally distributed. This is very convenient for statistical analy
5
6       Unfortunately stock returns are not normally distributed. You can described the reason in a variety of ways
7
8       However, there is a fair amount of evidence that stock returns are approximately normally distributed if you
9       To do this is fairly simple.
10
11
12      Below is the data for IBM from Feb, 1998 to May, 2003
The blue numbers are the ln returns which are in general the ones you want to use for analysis such as com
13      Beta's and it is also the data needed to implement the Option Pricing models.

14        Date       Close     Dividend          (Pt+Dt)/Pt-1      ln(1+Return)
15                                                                                  #NAME?
16        Feb-98       51.81
17        Mar-98       51.53                             0.99460          -0.542%   #NAME?
18        Apr-98       57.48                             1.11547          10.927%
19        May-98       58.29                             1.01409           1.399%
20        Jun-98       56.96                             0.97718          -2.308%
21         Jul-98      65.73                             1.15397          14.321%
22        Aug-98       55.87                             0.84999         -16.253%
23        Sep-98       63.75                             1.14104          13.194%
24        Oct-98       73.67                             1.15561          14.463%
25        Nov-98       81.92                             1.11199          10.615%
26        Dec-98       91.47                             1.11658          11.027%
27        Jan-99       90.91                             0.99388          -0.614%
28        Feb-99       84.21                             0.92630          -7.656%
29        Mar-99       87.93                             1.04418           4.323%
30        Apr-99      103.78                             1.18026          16.573%
31        May-99      115.09                             1.10898          10.344%
32        Jun-99      128.24                             1.11426          10.819%
33         Jul-99     124.71                             0.97247          -2.791%
34        Aug-99      123.59                             0.99102          -0.902%
35        Sep-99      120.06                             0.97144          -2.898%
36        Oct-99       97.48                             0.81193         -20.834%
37        Nov-99      102.26                             1.04904           4.787%
38        Dec-99      107.03                             1.04665           4.559%
39        Jan-00      111.37                             1.04055           3.975%
40        Feb-00      101.95                             0.91542          -8.838%
41        Mar-00      117.45                             1.15204          14.153%
42        Apr-00      110.63                             0.94193          -5.982%
43        May-00      106.47                             0.96240          -3.833%
44        Jun-00      108.71                             1.02104           2.082%
45         Jul-00     111.37                             1.02447           2.417%
46        Aug-00      130.99                             1.17617          16.226%
47        Sep-00      111.74                             0.85304         -15.895%
48   Oct-00       97.73                0.87462    -13.397%
49   Nov-00       92.77                0.94925     -5.209%
50   Dec-00       84.34                0.90913     -9.527%
51   Jan-01      111.13                1.31764     27.584%
52   Feb-01       99.12                0.89193    -11.437%
53   Mar-01       95.43                0.96277     -3.794%
54   Apr-01      114.24                1.19711     17.991%
55   May-01      110.93                0.97103     -2.940%
56   Jun-01      112.61                1.01514      1.503%
57    Jul-01     104.39                0.92700     -7.580%
58   Aug-01       99.17                0.95000     -5.130%
59   Sep-01          91                0.91762     -8.598%
60   Oct-01      107.23                1.17835     16.412%
61   Nov-01      114.69                1.06957      6.726%
62   Dec-01      120.02                1.04647      4.543%
63   Jan-02      107.05                0.89193    -11.436%
64   Feb-02       97.36                0.90948     -9.488%
65   Mar-02      103.19                1.05988      5.816%
66   Apr-02       83.11                0.80541    -21.641%
67   May-02       79.82                0.96041     -4.039%
68   Jun-02       71.44                0.89501    -11.092%
69    Jul-02      69.85                0.97774     -2.251%
70   Aug-02       74.96     0.15       1.07530      7.260%
71   Sep-02       57.98                0.77348    -25.686%
72   Oct-02        78.5                1.35392     30.300%
73   Nov-02       86.59     0.15       1.10497      9.982%
74   Dec-02       77.21                0.89167    -11.466%
75   Jan-03       77.91                1.00907      0.903%
76   Feb-03       77.81     0.15       1.00064      0.064%
77   Mar-03       78.29                1.00617      0.615%
78   Apr-03       84.74                1.08239      7.917%
79   May-03       86.18                1.01699      1.685%

80
81             Monthly Average Return =           0.817%     #NAME?
82             Monthly Standard Deviation =      11.317%     #NAME?
83
84             Annual Average Return =            9.800%     #NAME?
85             Annual Standard Deviation =       39.203%     #NAME?
86
87
H           I

y convenient for statistical analysis.

d the reason in a variety of ways but the basic reason is that stock prices cannot be negative.

mately normally distributed if you take the log (natural) of the return.

t to use for analysis such as computing
ls.
Other applications of logarithms: Trends
ROW/COL                B             C   D           E         F        G          H            I
5      Other applications of logarithms
6
7      Suppose you have the following series of numbers for which you are trying to forecast the next one
8                     10%
9      Year     Sales     Year       ln(sales)
10             1 \$100.00            1     4.6052 #NAME?
11             2 \$110.00            2     4.7005 #NAME?
12             3 \$121.00            3     4.7958 #NAME?
13             4 \$133.10            4     4.8911 #NAME?
14             5 \$146.41            5     4.9864 #NAME?
15             6 \$161.05            6     5.0817 #NAME?
16             7 \$177.16            7     5.1770 #NAME?
17             8 \$194.87            8     5.2723 #NAME?
18             9 \$214.36            9     5.3677 #NAME?
19            10 \$235.79           10     5.4630 #NAME?
20            11 \$259.37           11     5.5583 #NAME?
21            12 \$285.31           12     5.6536 #NAME?
22            13     ?             13      ?

23      The linear trend forecasts:
24      Forecast Series                         ln(series) Exp(ln forecast)
25             13 \$286.25                  13       5.7489 \$313.84 #NAME?

26      #NAME?                                             #NAME?
27
28      It is clear that the process is not linear. Thus a linear estimate would not be expected to produce good estim
29
30
31
32                                                  Sales
33
34
\$350.00                                                                              6.0000
35
36                                                                            Forecast                        5.8000
\$300.00
37                                                                                                            5.6000
38
\$250.00                                                                              5.4000
39
Sales (\$)

40                                                                                                            5.2000
ln(Sales)

\$200.00
41
5.0000
42                       \$150.00                                    Trend Line
43                                                                                                            4.8000
44                       \$100.00                                                                              4.6000
45
Raw Data                                                      4.4000
46                        \$50.00
47                                                                                                            4.2000
48                         \$0.00                                                                              4.0000
49                                 0               5                 10                  15
0   5          10   15
50
Time
51
52
53
54
ast the next one

d to produce good estimate

Natural Log of Sales

6.0000
Forecast
5.8000
5.6000
5.4000
5.2000
Trend Line
5.0000
4.8000
4.6000
4.4000
4.2000
4.0000
0                 5                10                  15
0   5          10   15
Time
Multiplicative Demand Functions
ROW/COL              B                   C             D          E          F          G
5      For demand equations we can also use the following:
6

8      This is called a multiplicative demand function.
9      The exponents are the elasticities of the variables.
10
11

12      Ln(Q) = a + c*ln(P) + d*ln(A) + e*ln(Y)
13
14
15      Some data:
16
17                 Q                   P            A           Y
18                          500         105.00      100.00     500.00
19                         1800          90.00      195.00     576.54
20                          900          75.00       95.00     600.00
21                         1400         130.00      145.00     706.23
22                         1800          59.00      120.00     750.00
23                         1200         114.00      169.00     810.00
24                         1300         105.00       95.00     795.00
25                          975         165.00      152.00     815.00
26                         2000         134.00      202.00     900.00
27
28      A linear fit using the above data
29

30      Q= f + g*P + h*A + i*Y
31
32
33      SUMMARY OUTPUT
34
35              Regression Statistics
36      Multiple R               0.888864171
37      R Square                 0.790079515
39      Standard Error           283.3424136
40      Observations                       9
41
42      ANOVA
43                                    df          SS        MS         F    Significance F
44      Regression                           3 1510807.61 503602.5 6.2728475   0.0379091
45      Residual                             5 401414.617 80282.92
46      Total                                8 1912222.22
47
48                                CoefficientsStandard Error t Stat     P-value   Lower 95%
49   Intercept                    -142.008306 572.243073 -0.24816 0.8138798 -1613.00355
50   X Variable 1                 -9.98905184 3.65624537 -2.73205 0.0411796 -19.3877144
51   X Variable 2                 8.198651835 2.7095159 3.025873 0.0292191 1.23363086
52   X Variable 3                 1.933355427 0.88602668 2.182051 0.0809034 -0.34424493
53
54   Estimating the results
55                                                                                           #NAME?
Estimated
56              Q                     P          A          Y            Q        Actual
57                       500           105.00    100.00    500.00           596        500
58                      1800            90.00    195.00    576.54          1672       1800
59                       900            75.00     95.00    600.00          1048        900
60                      1400           130.00    145.00    706.23          1114       1400
61                      1800            59.00    120.00    750.00          1702       1800
62                      1200           114.00    169.00    810.00          1671       1200
63                      1300           105.00     95.00    795.00          1125       1300
64                       975           165.00    152.00    815.00          1032        975
65                      2000           134.00    202.00    900.00          1916       2000
66
67
68   Now try the non-linear fit
69
70   Take the logarithms of the data
71
72            ln(Q)                 ln(P)       ln(A)      ln(Y)
73                     6.2146          4.6540     4.6052    6.2146
74                     7.4955          4.4998     5.2730    6.3570
75                     6.8024          4.3175     4.5539    6.3969
76                     7.2442          4.8675     4.9767    6.5599
77                     7.4955          4.0775     4.7875    6.6201
78                     7.0901          4.7362     5.1299    6.6970
79                     7.1701          4.6540     4.5539    6.6783
80                     6.8824          5.1059     5.0239    6.7032
81                     7.6009          4.8978     5.3083    6.8024
82
83   The multiplicative fit
84
85   SUMMARY OUTPUT
86
87           Regression Statistics
88   Multiple R               0.889270086
89   R Square                 0.790801287
91   Standard Error           0.251221433
92   Observations                       9
93
94   ANOVA
95                                   df          SS       MS         F    Significance F
96   Regression                            3 1.19286622 0.397622 6.3002402 0.03759558
97   Residual                              5 0.31556104 0.063112
98   Total                                 8 1.50842726
99
100                             CoefficientsStandard Error t Stat P-value Lower 95%
101   Intercept                  -1.81142129 3.01281415 -0.60124 0.573897 -9.55609396
102   X Variable 1               -0.84307889 0.32376373 -2.60399 0.048017 -1.67533869
103   X Variable 2              0.922450092 0.34014869 2.711902 0.0421803 0.04807146
104   X Variable 3              1.266559782 0.51433052 2.462541 0.0570503 -0.05556676
105
106
107
108   Remember that the above results are in logs, so the answer will be the log of sales and you would
109   need to take the anti-log (exp()) of the number ot get the resulting quantity sold
110
111                                                                    #NAME?
Estimated Estimated
112            ln(Q)               ln(P)       ln(A)       ln(Y)     ln(Q)       exp(Q)
113                    6.2146         4.6540     4.6052     6.2146        6.3841         592
114                    7.4955         4.4998     5.2730     6.3570        7.3105        1496
115                    6.8024         4.3175     4.5539     6.3969        6.8514         945
116                    7.2442         4.8675     4.9767     6.5599        6.9842        1079
117                    7.4955         4.0775     4.7875     6.6201        7.5518        1904
118                    7.0901         4.7362     5.1299     6.6970        7.4099        1652
119                    7.1701         4.6540     4.5539     6.6783        6.9242        1017
120                    6.8824         5.1059     5.0239     6.7032        7.0081        1106
121                    7.6009         4.8978     5.3083     6.8024        7.5716        1942
122
123                                                                               #NAME?
124
125
H       I

gnificance F

Upper 95%
1328.986939
-0.590389268
15.16367281
4.210955787

#NAME?

Error
96 #NAME?
-128
148
-286
-98
471
-175
57
-84

gnificance F
Upper 95%
5.933251388
-0.010819089
1.796828719
2.588686326

of sales and you would
sold

Actual          error
500             92 #NAME?
1800           -304
900             45
1400           -321
1800            104
1200            452
1300           -283
975            131
2000            -58

```
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