[Math] Deriving a power equation from a log-log line equation

logarithms

I have a log-log plot of my data (see below)

enter image description here

The equation of the line was determined to be: $5.26 + x0.7089$

If I wanted to convert this into a power equation would the correct way be:

$ ln(y) = A + B ln(x) $

Taking the antilog of both sides will give

y = $(e^A)(x^B)$

Let $e^A$ = A, and you have

$y = a(x^B)$

Is this the correct way to convert a line equation of log-log plot into a power equation in the form of $Ax^B$

If this is correct, why isnt the anti-log taken i.e. 10^5.26 instead of e^5.26

Best Answer

Supposing that both axis use b-basis logarithms we have : $$\log_b(y)=A+B\,\log_b(x)$$ using $\ b^{\log_b(y)}=y,\ b^{c+d}=b^c\,b^d\ $ and $\ B\,\log_b(x)=\log_b(x^B)\ $ we get : $$y=b^A\, x^B$$ So that your $\,y=e^A\,x^B$ equality was right for the natural logarithm $\,\ln=\log_e\,$ while the antilog expression ($y=10^{5.26}\,x^{0.7089}$) was the correct choice for the $10$-basis logarithm $\log_{10}$.

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