Here is a method using scaling.  The tangent is drawn at x = 0.8.

    f[t_] := (1 - t)^(-0.5);
    Print@LogPlot[f[t], {t, 0, 1},
       PlotRange -> {{0, 1}, {0.66, 20}},
       Frame -> True, PlotLabel -> "Original plot",
       AspectRatio -> 1];
    rescale[i_] := Exp[(i - 1) Log[2.]];
    Print[Column[{"Rescaling function: Exp[(i-1) Log[2.]]",
        Row[{"E.g. ", Table[{i, rescale[i]}, {i, 0, 4}]}]}]];
    normLog[n_] := 1 + Log[n]/Log[2.];
    normf[t_] := normLog[f[t]];
    Print@Plot[normf[t], {t, 0, 1},
       PlotRange -> {{0, 1}, normLog /@ {0.66, 20}},
       Frame -> True, PlotLabel -> "Rescaled plot",
       AspectRatio -> 1];
    ndf = D[normf[t], t];
    x = 0.8;
    m = ndf /. t -> x;
    y = normf[x];
    c = y - m x;
    x2 = x - 0.18;
    y2 = m x2 + c;
    x3 = x + 0.18;
    y3 = m x3 + c;
    newticks = {#, rescale[#]} & /@ Range[6];
    Print@Show[Plot[normf[t], {t, 0, 1},
        PlotRange -> {{0, 1}, normLog /@ {0.66, 20}},
        Frame -> True, PlotLabel -> "Plot with tangent",
        AspectRatio -> 1],
       Graphics[Arrow[{{x2, y2}, {x, y}, {x3, y3}}]],
       FrameTicks -> {Automatic, newticks}];

![enter image description here][1]


  [1]: https://i.stack.imgur.com/vowaK.png