Exponential models S1 question

Data are coded using y=Log y and X=Log x to give a linear relationship. The equation of the regression line for the coded data is Y=1.2 + 0.4X

a) State whether the relationship between y and x is in the form ax^n or y=kb^x.

Data are coded using y=Log y and X=Log x to give a linear relationship. The equation of the regression line for the coded data is Y=1.2 + 0.4X

a) State whether the relationship between y and x is in the form ax^n or y=kb^x.

Thoughts??

Not a clue if I'm honest.

Sub in Y = log(y) and X = log(x) and reareange for y. What form do you get??

I still don't understand. I rearrange to get y=10^1.2 . 10^0.4logx

Yeah, now notice that $10^{0.4\log x} = 10^{\log x^{0.4}} = ??$

What's the last equality?

x=10^0.4? I'm really confused.

Why does that first line give x^0.4?

Because $10^{\log x} = \log(10^x) = x$

Log and exponential are inverse operations of each other. They cancel out.

Could you just use the power rule i.e. logx.10 = log10x = x? Or is that wrong?

Well by the power rule we have $\log(10^x) =x \log(10) = x$

But to show that $10^{\log x} =x$ we just denote $y = 10^{\log{x}}$, take logs to get $\log y = \log(10^{\log{x}}) = \log x$ then this is only possible if $x=y$, hence we have that $x = 10^{\log{x}}$.