# In science proving something by making it linear is a common?

## Main Question or Discussion Point

In science proving something by making it linear is a cmmon approach.

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?

That does not represent a linear function of x.

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

HallsofIvy
Homework Helper
If you let X= log(x), then y= aX+ b is certainly linear. I imagine that is what you are thinking of. If you have data that does not appear to fall on a straight line when you plot (x,y), plotting (X, y) with X= log(x) may help. Sometimes plotting (log(x), log(y)) will help.

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.

D H
Staff Emeritus
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 $y(x)$, the linear approximation for $x\approx x_0$ is

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

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

$$y(x) \approx a \log x_0 + b + a\frac {x-x_0}{x_0} = \frac a {x_0} x + a(\log x_0-1) + b$$

Linearizing around $x_0=1$ yields a very simple result:

$$y(x) \approx a (x-1) + b$$

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