Logs Essay

596 WordsApr 4, 20123 Pages
Exponential and Logarithmic Models Typically, data is modelled either with a polynomial, power, periodic, exponential or logarithmic function. For example: Polynomial: y=ax2+bx+c (or y=ax-h2+k); y=ax3+bx2+cx+d; Periodic (i.e. sinusoidal): y=asin⁡[bx+c]+d (or y=asinbx+c+d) Exponential: y=a×eb(x+c)+d or y=a×bc(x+d)+e Logarithmic: y=alnbx+c+d; y=alog10bx+c+d Note that with the exception of a linear or quadratic equation, these all involve 4 parameters, typically a, b, c & d. These parameters usually refer to: * a vertical dilation (generally a); * a vertical translation (generally d) * a horizontal dilation (generally 1b) and * a horizontal translation (generally - c) of the basic function: y=x2; y=sinx; y=ex; y=lnx. In addition to these, your Classpad can fit a power function y=axb or a logistic function y=c1+a×e-bx. In the case of quadratic, exponential and logarithmic functions, one of the parameters is superfluous. Since y=ax2 and y=ax2 are the same function (i.e. have the same graph), a vertical dilation of a is the same as a horizontal dilation of 1a . So for a quadratic function a horizontal dilation is unnecessary. Using the laws of indices: y=a×ebx+c+d =a×ebx+bc+d . =a×ebx×ebc+d =f×ebx+d [where f=a×ebc] So a horizontal translation c is unnecessary. Using the laws of logarithms: y=alog10bx+c+d =alog10b+log10x+c+d =alog10b+alog10x+c+d =alog10x+c+f [where f=alog10b+c So a horizontal dilation b is not necessary. When your Classpad does a regression analysis, it calculates all three parameters for a quadratic function and all four parameters for a cubic or sinusoidal function. A power regression resembles a polynomial function in which all terms except for the leading term are assumed to have zero coefficients, i.e. all the parameters apart from a are zero. This will usually give a fairly poor

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