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My maths is not very good, so please bare with me if the answer is obvious. I have a log-log plot below, which I have generated in R.

enter image description here

I now need to find what i believe to be called the inverse to the function of the plot, resulting in me obtaining the following equation:

$y = constant . x^m$

I understand that $m$ will be the gradient of the line (0.71), but how do I calculate the constant?

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1 Answer 1

up vote 2 down vote accepted

CORRECTION (the constant factor depends of the $\log$ basis considered!) :

If $\log$ means $\log_{10}$ on both axis then

you got $\ \log_{10}(y)=5.3+0.71\log_{10}(x)\ $ (since $\ \log_{10}\bigl(x^{0.71}\bigr)=0.71\;\log_{10}(x)$).

Take the 'antilog' $10^x$ to obtain $\ y\approx 10^{5.3}\ x^{0.71}$ or : $$\boxed{y\approx 200000\ x^{0.71}}$$ (because $\log_{10}(2)\approx 0.3$ and with $2$ digits precision)

else if the 'natural logarithm' i.e. $\ln$ was taken on both sides then :

You got $\ \ln(y)=5.3+0.71\ln(x)$.

take the exponential of this to obtain $\ y\approx e^{5.3}\ x^{0.71}$ or : $$\boxed{y\approx 200\ x^{0.71}}$$

(because $\,e^{a+b}=e^a e^b\,$ and $\,x^c= e^{c\ln(x)}$)

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