- #1

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For instance you can write some laws on this form.

y=a*log(x)+b

But there lies my problem, because I cannot see how this should represent a linear line. So please does this represent a line? ANd if yes how come?

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- Thread starter lonerider
- Start date

- #1

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For instance you can write some laws on this form.

y=a*log(x)+b

But there lies my problem, because I cannot see how this should represent a linear line. So please does this represent a line? ANd if yes how come?

- #2

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That does not represent a linear function of x.

- #3

Gib Z

Homework Helper

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Perhaps a is of the form 1/(n log x)...>.<"

- #4

HallsofIvy

Science Advisor

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Stationery stores used to sell "semi-log graph paper" and "log-log graph paper" that already had one or both axes marked in term of the logarithm so that the conversion was automatic.

- #5

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Are you thinking of a linear approximation? Scientists and engineers use frequently use a linear approximation to make a problem more amenable to analysis.

In general, one makes a linear approximation to some function about some central point as the first-order Taylor expansion of the function at that point. For a one-dimensional function [itex]y(x)[/itex], the linear approximation for [itex]x\approx x_0[/itex] is

[tex]y(x) \approx y(x_0) + (x-x_0)\left.\frac{dy}{dx}\right|_{x=x_0}[/tex]

For your simple example, [itex]y=a\log x + b[/itex], the linear approximation for [itex]x\approx x_0[/itex] is

[tex] y(x) \approx a \log x_0 + b + a\frac {x-x_0}{x_0} = \frac a {x_0} x + a(\log x_0-1) + b[/tex]

Linearizing around [itex]x_0=1[/itex] yields a very simple result:

[tex] y(x) \approx a (x-1) + b[/tex]

In general, one makes a linear approximation to some function about some central point as the first-order Taylor expansion of the function at that point. For a one-dimensional function [itex]y(x)[/itex], the linear approximation for [itex]x\approx x_0[/itex] is

[tex]y(x) \approx y(x_0) + (x-x_0)\left.\frac{dy}{dx}\right|_{x=x_0}[/tex]

For your simple example, [itex]y=a\log x + b[/itex], the linear approximation for [itex]x\approx x_0[/itex] is

[tex] y(x) \approx a \log x_0 + b + a\frac {x-x_0}{x_0} = \frac a {x_0} x + a(\log x_0-1) + b[/tex]

Linearizing around [itex]x_0=1[/itex] yields a very simple result:

[tex] y(x) \approx a (x-1) + b[/tex]

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