What is / are the major properties of a "log log plot"?

Question

The only log chart I have tried before was a log scale on the $y$-axis, so that exponential growth of a number (such as stock prices) on the $y$-axis becomes a straight line.

But today I know there is something called a "$\log \log$-plot", meaning both the x and y axis has a log scale.

(Edit: the following about polynomial is not correct. It is only true for "power functions". See the info in the answers). It seems the plot can make any "polynomial function"

$$ y = a_0 x^n + a_1 x^{n-1} + ... + a_n, $$

become a straight line, probably with the slope of a number $a_0n + a_1(n-1) + a_2(n-2) + \ldots $

such as

$$ y = 8x^3 + 7x^2 + 6x + 5, $$

then the slope of the line is $8 \times 3 + 7 \times 2 + 6 \times 1 + 0 = 44$ on the $\log \log$-plot. (It feels weird that the $ x^3 $ is usually seen as powerful, but the 8 makes the slope much bigger than the effect of the $ x^3 $). Are there other use or major properties of a $\log \log$-plot?

Answer 1

A double-log coordinate system makes power functions: $$ y = ax^b $$ become straight lines. The parameter $b$ determines the visual slope on the line; $a$ determines its location (up/down).

Taking the logarithm on both sides of $y=ax^b$ gives you $$\log y = \log a + b\log x.$$ You can now set $y'=\log y$ and $x'=\log x$ for physical plotting distances and get $$y' = bx' + \log a$$

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Polynomials with more than one term are not straight lines in such a coordinate system. (But they have straight asymptotes corresponding to their first and last nonzero terms).

Answer 2

This isn't right though, OP. Given $y=p(x)$ where $p$ is a polynomial, the asymptotic slope of the log-log plot is the degree of $p$; the lower-order terms vanish in their effect on the log-log plot for $x$ large. The log-log plot of $y=8x^3$ would have a slope of 3; e.g., $\log y =3 \log x + \log 8$. The log-log plot of $y=8x^3+7x^2+6x+5$ would be a curve above the line $\log y = 3 \log x+ \log 8$ but would converge to this line for large $x$.

In fact, given $y = a_rx^r + \sum_{i=0}^{r-1}a_ix^i$, the log-log plot of $y$ versus $x$ would converge for large $x$ to the line $\log y = r \log x+ \log a_r$.