Intuition behind log plotting

Question

The two plots are the same, except the 2nd one has been log transformed on the y axis.

Could I please draw your attention to what happens when Theta >0.65? After taking the log-lin plot it seems that the two lines are constant relative to each other. Is there nothing more interesting to say about this other than that it is due to the difference in y axis scaling?

Any help would be appreciated!

The 3rd plot is plot(theta, [log(blue)-log(green)])

Additional Note: The green line is derived from the Blue line with this relationship:

Green = (0.95)*Blue^Theta

Answer 1

When the difference between curves on a logarithmic scale looks constant, it means that one curve is scaled by some power of 10 (or whatever the logarithmic base is) from the other curve.

The difference in the curves can be seen as $\Delta_{\log} = \log y_1 - \log y_2$. Of course, using properties of logarithms, this is just $$\Delta_{\log} = \log y_1 - \log y_2 = \log \frac{y_1}{y_2}.$$

Expanding this, you see that $$\frac{y_1}{y_2} = 10^{\Delta_{\log}},$$

or in other words,

$$y_1 = 10^{\Delta_{\log}}y_2.$$

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The behavior on your lin-lin plot after $\theta > 0.65$ is because $y_2$, the slope of the green line, is large enough that the slope of the blue line gets exaggerated as well.

Answer 2

You stated the relation between Blue and Green as $$ G = 0.95 ~ B ^ \Theta $$ semilogy shows data on the abszissa on a log$_{10}$-scale. Since $$ \log_{10}(G) = \log_{10}(0.95) + \log_{10}(B) ~ \Theta $$ the difference is $$ \log_{10}(B) - \log_{10}(G) = \log_{10}(B) ~ (1 - \Theta) - \log_{10}(0.95). $$ This is clearly not constant, it depends on $\Theta$ directly and through Blue.

From your difference plot one can also see that the factor Blue / Green varies, from about 3 to about 20, or the log$_{10}$-distance from log$_{10}$(3) = 0.477 to log$_{10}$(20) = 1.301. Not a large change, but again, not constant.

The circumstance that the lines appear to be approximately parallel must have to do a) with the special form of the function Blue, and b) with the fact that our eye can't reliable judge the distance of sloped and curved lines.