Fitting plot and data to an equation

If I have the following data:

data={{2, 66.7635}, {Log[300]/Log[10], 69.9679}, {Log[600]/Log[10],    71.54}, {3, 72.2428}, {-2.30103, 54.0023}, {-(Log[60]/Log[10]),    55.1941}, {-(Log[20]/Log[10]), 56.0038}, {-1,    56.9497}, {-(Log[6]/Log[10]), 57.305}, {-(Log[10/3]/Log[10]),    57.7213}, {-(Log[2]/Log[10]), 58.2489}, {-2.30103,    54.0367}, {-(Log[60]/Log[10]), 55.1157}, {-(Log[20]/Log[10]),    56.1704}, {-1, 56.7117}, {-(Log[6]/Log[10]),    57.2506}, {-(Log[10/3]/Log[10]), 57.7097}, {-(Log[2]/Log[10]),    58.1068}} 

Which looks like this plotted:

enter image description here

I have two questions:

1) How can I fit and plot the fit of this data based on the following equation?

: `` where Tf'_ref=57.2506 , q_ref=0.166667 and c1 and c2 are the fitting parameters. Also, notice that data is Tf' vs Log q in the equation.

2) How can I find the values of c1 and c2 which are the fitting parameters.

The fitting (orange line) is supposed to look like this (done in excel):

enter image description here

EDIT: I tried using NonLinearFitModel like this: Table[{NonlinearModelFit[data, Logqref - ((c1*(data[[i, 2]] - Tfref))/(c2*(data[[i, 2]] - Tfref))), {{c1, 8.6}, {c2, 17.2}}, x]; }, {i, 1, 11}] but this does not work. The reason I tried this is because data[[i, 2]] represents Tf' in the equation. Here Logref=Log10[0.16667]