I am trying to fit some data points to a transformed natural log function and transformed 1/x function. However, the fitted 1/x function fails to adhere nicely to the data points when x is growing larger. Below is my code:
pHBarListChurakov = {10.396067415730338`, 10.497191011235955`, 10.547752808988765`, 10.598314606741573`, 10.69943820224719`, 10.851123595505618`, 10.901685393258427`, 11.398876404494382`, 11.601123595505618`, 11.803370786516854`, 11.997191011235955`, 12.19943820224719`, 12.25`, 12.401685393258427`, 12.603932584269662`}; rhoOHBarList = 10^(pHBarListChurakov - 11); calculatedUList = {-0.9000016984445702`, -0.5617336372971362`, \ -0.9009333610974087`, -0.40213804143299425`, -0.05513091822567246`, 0.5116416122750036`, 0.2772878238568564`, 2.0185896240329964`, 2.721886178225983`, 3.4101441103082335`, 4.350952842429438`, 5.258892468165652`, 5.84833335802726`, 6.659730475621987`, 6.6045283567870126`}; fitModel = k1/(x(*-k2*)) - k3; fitCoeff = FindFit[Transpose[{rhoOHBarList, calculatedUList}], {fitModel}, {k1(*, k2*), k3}, x] fittedUFunc[x_] = fitModel /. fitCoeff; fitModel2 = k1*Log[k3*x] + k2; fitCoeff2 = FindFit[Transpose[{rhoOHBarList, calculatedUList}], {fitModel2, k3 > 0}, {k1, k2, k3}, x] fittedUFunc2[x_] = fitModel2 /. fitCoeff2; oriPlot = ListPlot[Transpose[{rhoOHBarList, calculatedUList}], PlotLabel -> "var vs Bulk OH Conc (mMol/L)"]; fittedPlot = Plot[fittedUFunc[x], {x, 0, 40}, PlotRange -> All, PlotLegends -> {"1/x"}]; fittedPlot2 = Plot[fittedUFunc2[x], {x, 0, 40}, PlotRange -> All, PlotLegends -> {"ln(x)"}, PlotStyle -> Orange]; Show[fittedPlot, fittedPlot2, oriPlot, PlotLabel -> "Calculated & Fitted u vs Bulk OH conc", AxesOrigin -> {0, 0}, PlotRange -> {-20, 7}] Here is the output: 
I want the transformed 1/x function to satisfy 2 requirements as shown in the picture.
