For this response, I will the base-2 logarithm. If you obtained the regression parameters

• slope: -1.12
• intercept: 3.2

after transforming $(x,y)$ to $(\log_2(x),\log_2(y))$, then we have the following model $$\log_2(y) = 3.2 - 1.12\log_2(x) + \epsilon$$ which can be rewritten as $$y = \frac{2^{3.2}\cdot2^\epsilon}{x^{1.12}}$$

The log-log transform is often called the power model because it estimates a power-relationship between the $x$ and $y$ variables.

With regards to fitting the untransformed data to the original points, you must take into account the $2^\epsilon$ factor which models the error of the estimation. In the linear regression, the residual standard error gives the spread of the error term, the distribution of $\epsilon$. It is centered at zero (which transforms to a multiplicative factor of $2^0=1$), and we assume a normal distribution. This means the "error" scaling factor would range from $2^{-2\epsilon}$ and $2^{2\epsilon}$ (for roughly the "middlemost" 95% of the values).

For this response, I will the base-2 logarithm. If you obtained the regression parameters

• slope: -0.116
• intercept: 1.19

after transforming $(x,y)$ to $(\log_2(x),\log_2(y))$, then we have the following model $$\log_2(y) = 1.19 - 0.116\log_2(x) + \epsilon$$ which can be rewritten as $$y = \frac{2^{1.19}\cdot2^\epsilon}{x^{0.116}}$$

The log-log transform is often called the power model because it estimates a power-relationship between the $x$ and $y$ variables.

With regards to fitting the untransformed data to the original points, you must take into account the $2^\epsilon$ factor which models the error of the estimation. In the linear regression, the residual standard error gives the spread of the error term, the distribution of $\epsilon$. It is centered at zero (which transforms to a multiplicative factor of $2^0=1$), and we assume a normal distribution. This means the "error" scaling factor would range from $2^{-2\epsilon}$ and $2^{2\epsilon}$ (for roughly the "middlemost" 95% of the values).