<< Graphics`Graphics`

a = (Sqrt[1 + 4*kτ*DoveruL]);

Clear[CAoverCA0]
CAoverCA0[DoveruL_, 
          kτ_] = ((4*a*(E^(1/2*1/DoveruL)))/((((1 + a))^2)*(E^(a/2*1/DoveruL)) - (((1 - a))^2)*(E^(-a/2*1/DoveruL))));

CApoverCA0[kτp_] = (E^(-kτp));

VoverVp = kτ/kτp;

Solve[1/(1 + 1) == CApoverCA0[kτp], kτp]

Solve[CAoverCA0[DoveruL, 1] == CApoverCA0[kτp], 
      kτp]

kτpatkτ1[
          DoveruL_] = -Log[-(4 (Sqrt[1 + 4 DoveruL]) (E^((1/(2 DoveruL)))))/((E^(-((Sqrt[1 + 4 DoveruL])/(2 DoveruL)))) - 
                    2 (Sqrt[1 + 4 DoveruL]) (E^(-((Sqrt[1 + 4 DoveruL])/(2 DoveruL)))) + (1 + 
                          4 DoveruL) (E^(-((Sqrt[1 + 4 DoveruL])/(2 DoveruL)))) - (E^(((Sqrt[1 + 4 DoveruL])/(2 DoveruL)))) - 
                    2 (Sqrt[1 + 4 DoveruL]) (E^(((Sqrt[1 + 4 DoveruL])/(2 DoveruL)))) - (1 + 
                          4 DoveruL) (E^(((Sqrt[1 + 4 DoveruL])/(2 DoveruL)))))];

middlekτ1Points = 
      Table[{N[CAoverCA0[(4^i), 1]], 
          1/N[kτpatkτ1[(4^i)]]}, {i, -2, 2}]

plugkτ1Point = Table[{N[(E^(-1))], 1}]

mixedkτ1Point = Table[{N[1/(1 + 1)], N[1/Log[2]]}]

kτ1Points = 
      Table[{plugkτ1Point, middlekτ1Points[[1]], 
          middlekτ1Points[[2]], middlekτ1Points[[3]], 
          middlekτ1Points[[4]], middlekτ1Points[[5]], 
          mixedkτ1Point}]

kτ1Line = 
        LogLogListPlot[kτ1Points, 
          PlotRange -> {{ .01, 1}, {1, 30}}, 
          GridLines -> {{ .01,  .02,  .03,  .04,  .05,  .06,  .07,  .08,  .09,  .1,  .2,  .3,  .4,  .5,  .6,  .7,  .8,  .9, 1}, {1, 2, 3, 4, 5, 6, 7, 
                8, 9, 10, 20, 30}}, 
          Ticks -> {{ .01,  .1, 1}, {1, 10, 30}}, 
          PlotJoined -> True, 
          PlotStyle -> Dashing[{ .02,  .015}]];

Solve[
      1/(1 + 2) == CApoverCA0[kτp], kτp]

Solve[CAoverCA0[DoveruL, 2] == CApoverCA0[kτp], 
      kτp]

kτpatkτ2[
          DoveruL_] = -Log[-(4 (Sqrt[1 + 8 DoveruL]) (E^((1/(2 DoveruL)))))/((E^(-((Sqrt[1 + 8 DoveruL])/(2 DoveruL)))) - 
                    2 (Sqrt[1 + 8 DoveruL]) (E^(-((Sqrt[1 + 8 DoveruL])/(2 DoveruL)))) + (1 + 
                          8 DoveruL) (E^(-((Sqrt[1 + 8 DoveruL])/(2 DoveruL)))) - (E^(((Sqrt[1 + 8 DoveruL])/(2 DoveruL)))) - 
                    2 (Sqrt[1 + 8 DoveruL]) (E^(((Sqrt[1 + 8 DoveruL])/(2 DoveruL)))) - (1 + 
                          8 DoveruL) (E^(((Sqrt[1 + 8 DoveruL])/(2 DoveruL)))))];

middlekτ2Points = 
      Table[{N[CAoverCA0[(4^i), 2]], 
          2/N[kτpatkτ2[(4^i)]]}, {i, -2, 2}]

plugkτ2Point = Table[{N[(E^(-2))], 1}]

mixedkτ2Point = Table[{N[1/(1 + 2)], N[2/Log[3]]}]

kτ2Points = 
      Table[{plugkτ2Point, middlekτ2Points[[1]], 
          middlekτ2Points[[2]], middlekτ2Points[[3]], 
          middlekτ2Points[[4]], middlekτ2Points[[5]], 
          mixedkτ2Point}]

kτ2Line = 
        LogLogListPlot[kτ2Points, 
          PlotRange -> {{ .01, 1}, {1, 30}}, 
          GridLines -> {{ .01,  .02,  .03,  .04,  .05,  .06,  .07,  .08,  .09,  .1,  .2,  .3,  .4,  .5,  .6,  .7,  .8,  .9, 1}, {1, 2, 3, 4, 5, 6, 7, 
                8, 9, 10, 20, 30}}, 
          Ticks -> {{ .01,  .1, 1}, {1, 10, 30}}, 
          PlotJoined -> True, 
          PlotStyle -> Dashing[{ .02,  .015}]];

Solve[
      1/(1 + 5) == CApoverCA0[kτp], kτp]

Solve[CAoverCA0[DoveruL, 5] == CApoverCA0[kτp], 
      kτp]

kτpatkτ5[
          DoveruL_] = -Log[-(4 (Sqrt[1 + 20 DoveruL]) (E^((1/(2 DoveruL)))))/((E^(-((Sqrt[1 + 20 DoveruL])/(2 DoveruL)))) - 
                    2 (Sqrt[1 + 20 DoveruL]) (E^(-((Sqrt[1 + 20 DoveruL])/(2 DoveruL)))) + (1 + 
                          20 DoveruL) (E^(-((Sqrt[1 + 20 DoveruL])/(2 DoveruL)))) - (E^(((Sqrt[1 + 20 DoveruL])/(2 DoveruL)))) - 
                    2 (Sqrt[1 + 20 DoveruL]) (E^(((Sqrt[1 + 20 DoveruL])/(2 DoveruL)))) - (1 + 
                          20 DoveruL) (E^(((Sqrt[1 + 20 DoveruL])/(2 DoveruL)))))];

middlekτ5Points = 
      Table[{N[CAoverCA0[(4^i), 5]], 
          5/N[kτpatkτ5[(4^i)]]}, {i, -2, 2}]

plugkτ5Point = Table[{N[(E^(-5))], 1}]

mixedkτ5Point = Table[{N[1/(1 + 5)], N[5/Log[6]]}]

kτ5Points = 
      Table[{plugkτ5Point, middlekτ5Points[[1]], 
          middlekτ5Points[[2]], middlekτ5Points[[3]], 
          middlekτ5Points[[4]], middlekτ5Points[[5]], 
          mixedkτ5Point}]

kτ5Line = 
        LogLogListPlot[kτ5Points, 
          PlotRange -> {{ .01, 1}, {1, 30}}, 
          GridLines -> {{ .01,  .02,  .03,  .04,  .05,  .06,  .07,  .08,  .09,  .1,  .2,  .3,  .4,  .5,  .6,  .7,  .8,  .9, 1}, {1, 2, 3, 4, 5, 6, 7, 
                8, 9, 10, 20, 30}}, 
          Ticks -> {{ .01,  .1, 1}, {1, 10, 30}}, 
          PlotJoined -> True, 
          PlotStyle -> Dashing[{ .02,  .015}]];

Solve[1/(1 + 10) == CApoverCA0[kτp], 
      kτp]

Solve[CAoverCA0[DoveruL, 10] == CApoverCA0[kτp], 
      kτp]

kτpatkτ10[
          DoveruL_] = -Log[-(4 (Sqrt[1 + 40 DoveruL]) (E^((1/(2 DoveruL)))))/((E^(-((Sqrt[1 + 40 DoveruL])/(2 DoveruL)))) - 
                    2 (Sqrt[1 + 40 DoveruL]) (E^(-((Sqrt[1 + 40 DoveruL])/(2 DoveruL)))) + (1 + 
                          40 DoveruL) (E^(-((Sqrt[1 + 40 DoveruL])/(2 DoveruL)))) - (E^(((Sqrt[1 + 40 DoveruL])/(2 DoveruL)))) - 
                    2 (Sqrt[1 + 40 DoveruL]) (E^(((Sqrt[1 + 40 DoveruL])/(2 DoveruL)))) - (1 + 
                          40 DoveruL) (E^(((Sqrt[1 + 40 DoveruL])/(2 DoveruL)))))];

middlekτ10Points = 
      Table[{N[CAoverCA0[(4^i), 10]], 
          10/N[kτpatkτ10[(4^i)]]}, {i, -2, 2}]

mixedkτ10Point = 
      Table[{N[1/(1 + 10)], N[10/Log[11]]}]

kτ10Points = 
      Table[{middlekτ10Points[[1]], middlekτ10Points[[2]], 
          middlekτ10Points[[3]], middlekτ10Points[[4]], 
          middlekτ10Points[[5]], mixedkτ10Point}]

kτ10Line = 
        LogLogListPlot[kτ10Points, 
          PlotRange -> {{ .01, 1}, {1, 30}}, 
          GridLines -> {{ .01,  .02,  .03,  .04,  .05,  .06,  .07,  .08,  .09,  .1,  .2,  .3,  .4,  .5,  .6,  .7,  .8,  .9, 1}, {1, 2, 3, 4, 5, 6, 7, 
                8, 9, 10, 20, 30}}, 
          Ticks -> {{ .01,  .1, 1}, {1, 10, 30}}, 
          PlotJoined -> True, 
          PlotStyle -> Dashing[{ .02,  .015}]];

Solve[1/(1 + 20) == CApoverCA0[kτp], kτp]

Solve[CAoverCA0[DoveruL, 20] == CApoverCA0[kτp], 
      kτp]

kτpatkτ20[
          DoveruL_] = -Log[-(4 (Sqrt[1 + 80 DoveruL]) (E^((1/(2 DoveruL)))))/((E^(-((Sqrt[1 + 80 DoveruL])/(2 DoveruL)))) - 
                    2 (Sqrt[1 + 80 DoveruL]) (E^(-((Sqrt[1 + 80 DoveruL])/(2 DoveruL)))) + (1 + 
                          80 DoveruL) (E^(-((Sqrt[1 + 80 DoveruL])/(2 DoveruL)))) - (E^(((Sqrt[1 + 80 DoveruL])/(2 DoveruL)))) - 
                    2 (Sqrt[1 + 80 DoveruL]) (E^(((Sqrt[1 + 80 DoveruL])/(2 DoveruL)))) - (1 + 
                          80 DoveruL) (E^(((Sqrt[1 + 80 DoveruL])/(2 DoveruL)))))];

middlekτ20Points = 
      Table[{N[CAoverCA0[(4^i), 20]], 
          20/N[kτpatkτ20[(4^i)]]}, {i, -2, 2}]

mixedkτ20Point = 
      Table[{N[1/(1 + 20)], N[20/Log[21]]}]

kτ20Points = 
      Table[{middlekτ20Points[[1]], middlekτ20Points[[2]], 
          middlekτ20Points[[3]], middlekτ20Points[[4]], 
          middlekτ20Points[[5]], mixedkτ20Point}]

kτ20Line = 
        LogLogListPlot[kτ20Points, 
          PlotRange -> {{ .01, 1}, {1, 30}}, 
          GridLines -> {{ .01,  .02,  .03,  .04,  .05,  .06,  .07,  .08,  .09,  .1,  .2,  .3,  .4,  .5,  .6,  .7,  .8,  .9, 1}, {1, 2, 3, 4, 5, 6, 7, 
                8, 9, 10, 20, 30}}, 
          Ticks -> {{ .01,  .1, 1}, {1, 10, 30}}, 
          PlotJoined -> True, 
          PlotStyle -> Dashing[{ .02,  .015}]];

Solve[1/(1 + 50) == CApoverCA0[kτp], 
      kτp]

Solve[CAoverCA0[DoveruL, 50] == CApoverCA0[kτp], 
      kτp]

kτpatkτ50[
          DoveruL_] = -Log[-(4 (Sqrt[1 + 200 DoveruL]) (E^((1/(2 DoveruL)))))/((E^(-((Sqrt[1 + 200 DoveruL])/(2 DoveruL)))) - 
                    2 (Sqrt[1 + 200 DoveruL]) (E^(-((Sqrt[1 + 200 DoveruL])/(2 DoveruL)))) + (1 + 
                          200 DoveruL) (E^(-((Sqrt[1 + 200 DoveruL])/(2 DoveruL)))) - (E^(((Sqrt[1 + 200 DoveruL])/(2 DoveruL)))) - 
                    2 (Sqrt[1 + 200 DoveruL]) (E^(((Sqrt[1 + 200 DoveruL])/(2 DoveruL)))) - (1 + 
                          200 DoveruL) (E^(((Sqrt[1 + 200 DoveruL])/(2 DoveruL)))))];

middlekτ50Points = 
      Table[{N[CAoverCA0[(4^i), 50]], 
          50/N[kτpatkτ50[(4^i)]]}, {i, -2, 2}]

mixedkτ50Point = 
      Table[{N[1/(1 + 50)], N[50/Log[51]]}]

kτ50Points = 
      Table[{middlekτ50Points[[1]], middlekτ50Points[[2]], 
          middlekτ50Points[[3]], middlekτ50Points[[4]], 
          middlekτ50Points[[5]], mixedkτ50Point}]

kτ50Line = 
        LogLogListPlot[kτ50Points, 
          PlotRange -> {{ .01, 1}, {1, 30}}, 
          GridLines -> {{ .01,  .02,  .03,  .04,  .05,  .06,  .07,  .08,  .09,  .1,  .2,  .3,  .4,  .5,  .6,  .7,  .8,  .9, 1}, {1, 2, 3, 4, 5, 6, 7, 
                8, 9, 10, 20, 30}}, 
          Ticks -> {{ .01,  .1, 1}, {1, 10, 30}}, 
          PlotJoined -> True, 
          PlotStyle -> Dashing[{ .02,  .015}]];

allkτs = 
        Show[kτ1Line, kτ2Line, kτ5Line, kτ10Line, 
          kτ20Line, kτ50Line];

Solve[
      CAoverCA0[ .0625, kτ] == CApoverCA0[kτp], 
      kτp]

kτpatDoveruL0625[
          kτ_] = -1.` Log[-(11923.831948166911` (Sqrt[1.` + 0.25` kτ]))/((E^(-8.`15.9546 (Sqrt[1 + 0.25`15.9546 kτ]))) - 
                    1.` (E^(8.`15.9546 (Sqrt[1 + 0.25`15.9546 kτ]))) - 
                    2.` (E^(-8.`15.9546 (Sqrt[1 + 0.25`15.9546 kτ]))) (Sqrt[1.` + 0.25` kτ]) - 
                    2.` (E^(8.`15.9546 (Sqrt[1 + 0.25`15.9546 kτ]))) (Sqrt[1.` + 0.25` kτ]) + (E^(-8.`15.9546 (Sqrt[1 + 0.25`15.9546 kτ]))) (1.` + 0.25` kτ) - 
                    1.` (E^(8.`15.9546 (Sqrt[1 + 0.25`15.9546 kτ]))) (1.` + 
                          0.25` kτ))];

DoveruL0625Points = 
      Table[{{1, 1}, {CAoverCA0[0.0625, 1], 
            1/kτpatDoveruL0625[1]}, {CAoverCA0[0.0625, 2], 
            2/kτpatDoveruL0625[2]}, {CAoverCA0[0.0625, 5], 
            5/kτpatDoveruL0625[5]}, {CAoverCA0[0.0625, 10], 
            10/kτpatDoveruL0625[10]}, {CAoverCA0[0.0625, 20], 
            20/kτpatDoveruL0625[20]}, {CAoverCA0[0.0625, 50], 
            50/kτpatDoveruL0625[50]}, {CAoverCA0[0.0625, 100], 
            100/kτpatDoveruL0625[100]}}]

DoveruL0625Line = 
        LogLogListPlot[DoveruL0625Points, 
          PlotRange -> {{ .01, 1}, {1, 30}}, 
          GridLines -> {{ .01,  .02,  .03,  .04,  .05,  .06,  .07,  .08,  .09,  .1,  .2,  .3,  .4,  .5,  .6,  .7,  .8,  .9, 1}, {1, 2, 3, 4, 5, 6, 7, 
                8, 9, 10, 20, 30}}, 
          Ticks -> {{ .01,  .1, 1}, {1, 10, 30}}, 
          PlotJoined -> True, 
          PlotStyle -> Thickness[ .005]];

Solve[
      CAoverCA0[ .25, kτ] == CApoverCA0[kτp], 
      kτp]

kτpatDoveruL025[
          kτ_] = -1.` Log[-(29.5562243957226` (Sqrt[1.` + 1.` kτ]))/((E^(-2.`15.9546 (Sqrt[1 + 1.`15.9546 kτ]))) - 
                    1.` (E^(2.`15.9546 (Sqrt[1 + 1.`15.9546 kτ]))) - 
                    2.` (E^(-2.`15.9546 (Sqrt[1 + 1.`15.9546 kτ]))) (Sqrt[1.` + 1.` kτ]) - 
                    2.` (E^(2.`15.9546 (Sqrt[1 + 1.`15.9546 kτ]))) (Sqrt[1.` + 1.` kτ]) + (E^(-2.`15.9546 (Sqrt[1 + 1.`15.9546 kτ]))) (1.` + 1.` kτ) - 
                    1.` (E^(2.`15.9546 (Sqrt[1 + 1.`15.9546 kτ]))) (1.` + 
                          1.` kτ))];

DoveruL025Points = 
      Table[{{1, 1}, {CAoverCA0[0.25, 1], 
            1/kτpatDoveruL025[1]}, {CAoverCA0[0.25, 2], 
            2/kτpatDoveruL025[2]}, {CAoverCA0[0.25, 5], 
            5/kτpatDoveruL025[5]}, {CAoverCA0[0.25, 10], 
            10/kτpatDoveruL025[10]}, {CAoverCA0[0.25, 20], 
            20/kτpatDoveruL025[20]}, {CAoverCA0[0.25, 50], 
            50/kτpatDoveruL025[50]}, {CAoverCA0[0.25, 100], 
            100/kτpatDoveruL025[100]}}]

DoveruL025Line = 
        LogLogListPlot[DoveruL025Points, 
          PlotRange -> {{ .01, 1}, {1, 30}}, 
          GridLines -> {{ .01,  .02,  .03,  .04,  .05,  .06,  .07,  .08,  .09,  .1,  .2,  .3,  .4,  .5,  .6,  .7,  .8,  .9, 1}, {1, 2, 3, 4, 5, 6, 7, 
                8, 9, 10, 20, 30}}, 
          Ticks -> {{ .01,  .1, 1}, {1, 10, 30}}, 
          PlotJoined -> True, 
          PlotStyle -> Thickness[ .005]];

Solve[
      CAoverCA0[1, kτ] == CApoverCA0[kτp], 
      kτp]

kτpatDoveruL1[
          kτ_] = -Log[-(4 (Sqrt[E]) (Sqrt[1 + 4 kτ]))/((E^(-(1/2) (Sqrt[1 + 4 kτ]))) - (E^((1/2) (Sqrt[1 + 4 kτ]))) - 
                    2 (E^(-(1/2) (Sqrt[1 + 4 kτ]))) (Sqrt[1 + 4 kτ]) - 
                    2 (E^((1/2) (Sqrt[1 + 4 kτ]))) (Sqrt[1 + 4 kτ]) + (E^(-(1/2) (Sqrt[1 + 4 kτ]))) (1 + 4 kτ) - (E^((1/2) (Sqrt[1 + 4 kτ]))) (1 + 4 kτ))];

DoveruL1Points = 
      Table[{{1, 1}, {N[CAoverCA0[1, 1]], 
            N[1/kτpatDoveruL1[1]]}, {N[CAoverCA0[1, 2]], 
            N[2/kτpatDoveruL1[2]]}, {N[CAoverCA0[1, 5]], 
            N[5/kτpatDoveruL1[5]]}, {N[CAoverCA0[1, 10]], 
            N[10/kτpatDoveruL1[10]]}, {N[CAoverCA0[1, 20]], 
            N[20/kτpatDoveruL1[20]]}, {N[CAoverCA0[1, 50]], 
            N[50/kτpatDoveruL1[50]]}, {N[CAoverCA0[1, 100]], 
            N[100/kτpatDoveruL1[100]]}}]

DoveruL1Line = 
        LogLogListPlot[DoveruL1Points, 
          PlotRange -> {{ .01, 1}, {1, 30}}, 
          GridLines -> {{ .01,  .02,  .03,  .04,  .05,  .06,  .07,  .08,  .09,  .1,  .2,  .3,  .4,  .5,  .6,  .7,  .8,  .9, 1}, {1, 2, 3, 4, 5, 6, 7, 
                8, 9, 10, 20, 30}}, 
          Ticks -> {{ .01,  .1, 1}, {1, 10, 30}}, 
          PlotJoined -> True, 
          PlotStyle -> Thickness[ .005]];

Solve[
      CAoverCA0[4, kτ] == CApoverCA0[kτp], 
      kτp]

kτpatDoveruL4[
          kτ_] = -Log[-(4 (E^(1/8)) (Sqrt[1 + 16 kτ]))/((E^(-(1/8) (Sqrt[1 + 16 kτ]))) - (E^((1/8) (Sqrt[1 + 16 kτ]))) - 
                    2 (E^(-(1/8) (Sqrt[1 + 16 kτ]))) (Sqrt[1 + 16 kτ]) - 
                    2 (E^((1/8) (Sqrt[1 + 16 kτ]))) (Sqrt[1 + 16 kτ]) + (E^(-(1/8) (Sqrt[1 + 16 kτ]))) (1 + 
                          16 kτ) - (E^((1/8) (Sqrt[1 + 16 kτ]))) (1 + 16 kτ))];

DoveruL4Points = 
      Table[{{1, 1}, {N[CAoverCA0[4, 1]], 
            N[1/kτpatDoveruL4[1]]}, {N[CAoverCA0[4, 2]], 
            N[2/kτpatDoveruL4[2]]}, {N[CAoverCA0[4, 5]], 
            N[5/kτpatDoveruL4[5]]}, {N[CAoverCA0[4, 10]], 
            N[10/kτpatDoveruL4[10]]}, {N[CAoverCA0[4, 20]], 
            N[20/kτpatDoveruL4[20]]}, {N[CAoverCA0[4, 50]], 
            N[50/kτpatDoveruL4[50]]}, {N[CAoverCA0[4, 100]], 
            N[100/kτpatDoveruL4[100]]}}]

DoveruL4Line = 
        LogLogListPlot[DoveruL4Points, 
          PlotRange -> {{ .01, 1}, {1, 30}}, 
          GridLines -> {{ .01,  .02,  .03,  .04,  .05,  .06,  .07,  .08,  .09,  .1,  .2,  .3,  .4,  .5,  .6,  .7,  .8,  .9, 1}, {1, 2, 3, 4, 5, 6, 7, 
                8, 9, 10, 20, 30}}, 
          Ticks -> {{ .01,  .1, 1}, {1, 10, 30}}, 
          PlotJoined -> True, 
          PlotStyle -> Thickness[ .005]];

Solve[
      CAoverCA0[16, kτ] == CApoverCA0[kτp], 
      kτp]

kτpatDoveruL16[
          kτ_] = -Log[-(4 (E^(1/32)) (Sqrt[1 + 64 kτ]))/((E^(-(1/32) (Sqrt[1 + 64 kτ]))) - (E^((1/32) (Sqrt[1 + 64 kτ]))) - 
                    2 (E^(-(1/32) (Sqrt[1 + 64 kτ]))) (Sqrt[1 + 64 kτ]) - 
                    2 (E^((1/32) (Sqrt[1 + 64 kτ]))) (Sqrt[1 + 64 kτ]) + (E^(-(1/32) (Sqrt[1 + 64 kτ]))) (1 + 
                          64 kτ) - (E^((1/32) (Sqrt[1 + 64 kτ]))) (1 + 64 kτ))];

DoveruL16Points = 
      Table[{{1, 1}, {N[CAoverCA0[16, 1]], 
            N[1/kτpatDoveruL16[1]]}, {N[CAoverCA0[16, 2]], 
            N[2/kτpatDoveruL16[2]]}, {N[CAoverCA0[16, 5]], 
            N[5/kτpatDoveruL16[5]]}, {N[CAoverCA0[16, 10]], 
            N[10/kτpatDoveruL16[10]]}, {N[CAoverCA0[16, 20]], 
            N[20/kτpatDoveruL16[20]]}, {N[CAoverCA0[16, 50]], 
            N[50/kτpatDoveruL16[50]]}, {N[CAoverCA0[16, 100]], 
            N[100/kτpatDoveruL16[100]]}}]

DoveruL16Line = 
        LogLogListPlot[DoveruL16Points, 
          PlotRange -> {{ .01, 1}, {1, 30}}, 
          GridLines -> {{ .01,  .02,  .03,  .04,  .05,  .06,  .07,  .08,  .09,  .1,  .2,  .3,  .4,  .5,  .6,  .7,  .8,  .9, 1}, {1, 2, 3, 4, 5, 6, 7, 
                8, 9, 10, 20, 30}}, 
          Ticks -> {{ .01,  .1, 1}, {1, 10, 30}}, 
          PlotJoined -> True, 
          PlotStyle -> Thickness[ .005]];

DoveruLinf[conc_] = (((1/conc) - 1)/Log)[(1/conc)]

DoveruLinfLine = 
        LogLogPlot[DoveruLinf[conc], {conc,  .01, 1}, 
          PlotRange -> {{ .01, 1}, {1, 30}}, 
          GridLines -> {{ .01,  .02,  .03,  .04,  .05,  .06,  .07,  .08,  .09,  .1,  .2,  .3,  .4,  .5,  .6,  .7,  .8,  .9, 1}, {1, 2, 3, 4, 5, 6, 7, 
                8, 9, 10, 20, 30}}, 
          Ticks -> {{ .01,  .1, 1}, {1, 10, 30}}, 
          PlotJoined -> True, 
          PlotStyle -> Thickness[ .005]];

allDoveruLs = 
        Show[DoveruL0625Line, DoveruL025Line, DoveruL1Line, DoveruL4Line, 
          DoveruL16Line, DoveruLinfLine];

Show[allkτs, allDoveruLs];