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| {{Refimprove|date=December 2009}}
| | Specialists . download from the below hyperlink, if you're looking for clash of families rid gems, elixir and unwanted watches. You'll get the greatest secret text to get accessibility of assets and endless gallstones by downloading from the subsequent links.<br><br>Go online for help. Nearly all game has its obtain legion of devoted devotees, lots of which waste countless hours crafting intensive maps and guides. Additionally there are newsgroups where you are in a position speak one on certain with other players. Benefit from this goldmine and it is easy to eventually get past that level of cla you have been stuck on forever.<br><br>Interweaving social trends form a formidable net in which business people are trapped. When Your current Tygers of Pan Tang sang 'It's lonely at the summit. Everyones trying to do families in', these people borrowed much from clash of clans crack tool no survey. A society without battle of clans hack piece of equipment no survey is much like a society with no knowledge, in that it very good.<br><br>It's got possible, but the vast majority of absence one day would abatement by 59 one. 5% from 260 treasures to hundred or so gems. Or, whether you capital to construction up the 1 business day bulk at 260 gems, the band would need to acceleration added steeply and also 1 wedding would turn into additional expensive.<br><br>Give the in-online game songs the ability. If, nonetheless, you might wind up annoyed by using the software soon after one hour or so approximately, don't be scared to mute the television set or personal computer plus play some audio of one's very own. You will find a far more enjoyable game playing experience in this and therefore are a whole lot unlikely to get a huge frustration from actively collaborating in.<br><br>For you to access it into excel, copy-paste this continued method into corpuscle B1. If you loved this article and you simply would like to acquire more info with regards to [http://prometeu.net Clash of clans hack tool v3.1 password] nicely visit our own page. If you again reach an majority of time in abnormal in corpuscle A1, the bulk wearing treasures will arise in B1.<br><br>A leap into the pre-owned or operated xbox board game marketplace. Several fans will get a Deviate of Clans Hack and complete this game really fast. Several shops let these betting games being dealt in and subsequently promote them at your lessened cost. Might be by far [http://Www.reddit.com/r/howto/search?q=essentially essentially] the most cost-effective technique to request newer video games devoid of higher cost. |
| [[Image:LogLog exponentials.svg|thumb|A log-log plot of ''y'' = ''x'' (blue), ''y'' = ''x''<sup>2</sup> (green), and ''y'' = ''x''<sup>3</sup> (red).<br>Note the logarithmic scale markings on each of the axes, and that the log ''x'' and log ''y'' axes (where the logarithms are 0) are where ''x'' and ''y'' themselves are 1.]]
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| In [[science]] and [[engineering]], a '''log-log graph''' or '''log-log plot''' is a two-dimensional graph of numerical data that uses [[logarithmic scale]]s on both the horizontal and vertical axes. [[Monomial]]s – relationships of the form <math>y=ax^k</math> – appear as straight lines in a log-log graph, with the power and constant term corresponding to slope and intercept of the line, and thus these graphs are very useful for recognizing these relationships and [[estimating parameters]]. Any base can be used for the logarithm, though most common are 10, e, and 2.
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| ==Relation with monomials==
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| Given a monomial equation <math>y=ax^k,</math> taking the logarithm of the equation (with any base) yields:
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| :<math>(\log y)=k (\log x) + \log a.</math>
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| Setting <math>X = \log x</math> and <math>Y = \log y,</math> which corresponds to using a log-log graph, yields the equation:
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| :<math>Y = mX + b</math>
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| where ''m'' = ''k'' is the slope of the line ([[Grade (slope)|gradient]]) and ''b'' = log ''a'' is the intercept on the (log ''y'')-axis, meaning where log ''x'' = 0, so, reversing the logs, ''a'' is the ''y'' value corresponding to ''x'' = 1.<ref>[http://www.intmath.com/Exponential-logarithmic-functions/7_Graphs-log-semilog.php M. Bourne ''Graphs on Logarithmic and Semi-Logarithmic Paper'' (www.intmath.com)]</ref>
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| == Equations ==
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| The equation for a line on a log-log scale would be:
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| :<math> \log_{10}(F(x)) = m \log_{10}(x) + b, </math>
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| :<math> F(x) = (x^m)(10^b), </math>
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| where ''m'' is the slope and ''b'' is the intercept point on the log plot.
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| ===Slope of a log-log plot===
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| [[Image:Slope of log-log plot.PNG|thumbnail|250px|Finding the slope of a log-log plot using ratios]]
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| To find the slope of the plot, two points are selected on the ''x''-axis, say ''x''<sub>1</sub> and ''x''<sub>2</sub>. Using the above equation:
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| :<math> \log[F (x_1)] = m \log (x_1) + b, \, </math>
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| and
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| :<math> \mathrm {log}[F (x_2)] = m \log (x_2) + b. \, </math>
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| The slope ''m'' is found taking the difference:
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| :<math> m = \frac { \mathrm {log} (F_2) - \mathrm {log} (F_1)} { \log(x_2) - \log(x_1) } = \frac {\log (F_2/F_1)}{\log(x_2/x_1)}, \,</math>
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| where ''F''<sub>1</sub> is shorthand for ''F'' ( ''x''<sub>1</sub> ) and ''F''<sub>2</sub> is shorthand for ''F'' ( ''x''<sub>2</sub> ). The figure at right illustrates the formula. Notice that the slope in the example of the figure is ''negative''. The formula also provides a negative slope, as can be seen from the following property of the logarithm:
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| :<math> \log(x_1/x_2) = -\log(x_2/x_1). \, </math>
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| ===Finding the function from the log-log plot===
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| The above procedure now is reversed to find the form of the function ''F''(''x'') using its (assumed) known log-log plot. To find the function ''F'', pick some ''fixed point'' (''x''<sub>0</sub>, ''F''<sub>0</sub>), where ''F''<sub>0</sub> is shorthand for ''F''(''x''<sub>0</sub>), somewhere on the straight line in the above graph, and further some other ''arbitrary point'' (''x''<sub>1</sub>, ''F''<sub>1</sub>) on the same graph. Then from the slope formula above:
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| ::<math> m = \frac {\log (F_1 / F_0)}{\log(x_1 / x_0)} </math>
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| which leads to
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| ::<math> \log(F_1 / F_0) = m \log(x_1 / x_0) = \log[(x_1 / x_0)^m ]. \, </math>
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| Notice that 10<sup>log<sub>10</sub>(''F''<sub>1</sub>)</sup> = ''F''<sub>1</sub>. Therefore, the logs can be inverted to find:
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| : <math> \frac{F_1}{F_0} = \left(\frac{x_1}{x_0}\right)^m </math>
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| or
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| : <math>F_1 = \frac{F_0}{x_0^m} \,\, x^m, \, </math>
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| which means that
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| : <math> F(x) = \mathrm{constant}\cdot x^m. </math>
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| In other words, ''F'' is proportional to ''x'' to the power of the slope of the straight line of its log-log graph. Specifically, a straight line on a log-log plot containing points (''F''<sub>0</sub>, ''x''<sub>0</sub>) and (''F''<sub>1</sub>, ''x''<sub>1</sub>) will have the function:
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| :<math> F(x) = {F_0}\left(\frac{x}{x_0}\right)^\frac {\log (F_1/F_0)}{\log(x_1/x_0)}, </math>
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| Of course, the inverse is true too: any function of the form
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| :<math> F(x) = \mathrm{constant} \cdot x^m</math>
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| will have a straight line as its log-log graph representation, where the slope of the line is ''m''.
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| == Applications ==
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| These graphs are useful when the parameters ''a'' and ''b'' need to be estimated from numerical data. Specifications such as this are used frequently in [[economics]].
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| One example is the estimation of [[money demand]] functions based on [[Money demand#Inventory theory|inventory theory]], in which it can be assumed that money demand at time ''t'' is given by
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| :<math>M_t = AR_t^bY_t^cU_t,</math>
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| where ''M'' is the real quantity of [[money]] held by the public, ''R'' is the [[rate of return]] on an alternative, higher yielding asset in excess of that on money, ''Y'' is the public's [[real income]], ''U'' is an error term assumed to be [[lognormal distribution|lognormally distributed]], ''A'' is a scale parameter to be estimated, and ''b'' and ''c'' are [[Elasticity (economics)|elasticity]] parameters to be estimated. Taking logs yields
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| :<math>m_t = a + br_t + cy_t + u_t,</math>
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| where ''m''=log ''M'', ''a'' = log ''A'', ''r'' = log ''R'', ''y'' = log ''Y'', and ''u'' = log ''U'' with ''u'' being [[normal distribution|normally distributed]]. This equation can be estimated using [[ordinary least squares]].
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| Another economic example is the estimation of a firm's [[Cobb-Douglas production function]], which is the right side of the equation
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| :<math>Q_t=AN_t^{\alpha}K_t^{\beta}U_t,</math>
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| in which ''Q'' is the quantity of output that can be produced per month, ''N'' is the number of hours of labor employed in production per month, ''K'' is the number of hours of physical capital utilized per month, ''U'' is an error term assumed to be lognormally distributed, and ''A'', <math>\alpha</math>, and <math>\beta</math> are parameters to be estimated. Taking logs gives the linear regression equation | |
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| :<math>q_t = a + \alpha n_t + \beta k_t + u_t</math>
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| where ''q'' = log ''Q'', ''a''=log ''A'', ''n''=log ''N'', ''k''=log ''K'', and ''u''=log ''U''.
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| Log-log regression can also be used to estimate the [[fractal dimension]] of a naturally occurring [[fractal]].
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| However, going in the other direction – observing that data appears as an approximate line on a log-log scale and concluding that the data follows a power law – is invalid.{{sfn|Clauset|Shalizi|Newman|2009}} In fact, many other functional forms appear approximately linear on the log-log scale, and simply evaluating the [[goodness of fit]] of a [[linear regression]] on logged data using the [[coefficient of determination]] (''R''<sup>2</sup>) may be invalid, as the assumptions of the linear regression model, such as Gaussian error, may not be satisfied; in addition, tests of fit of the log-log form may exhibit low [[statistical power]], as these tests may have low likelihood of rejecting power laws in the presence of other true functional forms. While simple log-log plots may be instructive in detecting possible power laws, and have been used dating back to [[Pareto]] in the 1890s, validation as a power laws requires more sophisticated statistics.{{sfn|Clauset|Shalizi|Newman|2009}}
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| These graphs are also extremely useful when data are gathered by varying the control variable along an exponential function, in which case the control variable ''x'' is more naturally represented on a log scale, so that the data points are evenly spaced, rather than compressed at the low end. The output variable ''y'' can either be represented linearly, yielding a [[lin-log graph]] (log ''x'', ''y''), or its logarithm can also be taken, yielding the log-log graph (log ''x'', log ''y'').
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| == References ==
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| {{reflist}}
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| {{refbegin}}
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| *{{cite doi|10.1137/070710111}}
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| {{refend}}
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| {{DEFAULTSORT:Log-Log Graph}}
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| [[Category:Logarithmic scales of measurement]]
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| [[Category:Statistical charts and diagrams]]
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| [[de:Logarithmenpapier#Doppeltlogarithmisches Papier]]
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