{Manipulate[
Show[
ParametricPlot3D[
{u Cos[v],u Sin[v],Log[u+(Sqrt[u u-1])]},{u,0,7},{v,0,2 Pi},PlotPoints-> 30,PlotStyle->Directive[Opacity[op]],ColorFunction->"TemperatureMap",
Mesh->5,MeshStyle-> Brown,Boxed->False,Axes-> False,ImageSize-> 400],

ParametricPlot3D[
{u Cos[Tan[ϕ]*Log[u+(Sqrt[u u-1])]+d[[2]]-Tan[ϕ]*Log[d[[1]]+(Sqrt[d[[1]]*d[[1]]-1])]],u Sin[Tan[ϕ]*Log[u+(Sqrt[u u-1])]+d[[2]]-Tan[ϕ]*Log[d[[1]]+(Sqrt[d[[1]]*d[[1]]-1])]],Log[u+(Sqrt[u u-1])]},{u,0,7},
PlotPoints-> 30,
Boxed->False,Axes-> False,PlotStyle->Directive[Blue,Thick]],

PlotLabel->Style["Локсодромия v = tg[ϕ] ln(u + (Sqrt[(u^2) - 1])) + const",Blue,15]],
{{ϕ,0,"Угол пересечения ϕ"},0,Pi/2-0.00001},
{{op,1,"Прозрачность"},0.2,1},
{{d,{1,0},"Начальная точка ( (Subscript[u, 0]), (Subscript[v, 0]))"},{1,0},{7,2*Pi}}

],

Manipulate[
Show[
ParametricPlot3D[{u Cos[v],u Sin[v],Log[u+(Sqrt[u u-1])]},{u,0,7},{v,0,2 Pi},PlotPoints-> 30,PlotStyle->Directive[Opacity[op]],Mesh->7,MeshStyle-> Brown,
Boxed->False,Axes-> False,ColorFunction->"TemperatureMap",ImageSize-> 500],ParametricPlot3D[{u0*Cos[v],u0 Sin[v],Log[u0+(Sqrt[u0*u0-1])]},{v,0,2*Pi},PlotPoints-> 30,Boxed->False,Axes-> False,PlotStyle->Directive[Blue,Thick]],PlotLabel->Style["Локсодромия, перпендикулярная кривой v=const, т.е. u=const",Blue,13]],{{u0,1,"Первая координата нач. точки (Subscript[u, 0])"},1,7},{{op,1,"Прозрачность"},0,1}]}