Additional tick marks in a LogLogPlot

Question

In the following mwe, I have a loglogplot. I want to display the tick positions where the horizontal and vertical lines meet the axes.

f[x_] = 10^(-10)*Log[x];

LogLogPlot[f[x], {x, 10^4, 10^7}, 
 Epilog -> { 
   Line[{{Log[5*10^6], Log[f[5*10^3]]}, {Log[5*10^6], 
      Log[f[5*10^6]]}}], 
   Line[{{Log[5*10^3], Log[f[5*10^6]]}, {Log[5*10^6], 
      Log[f[5*10^6]]}}]}]

I found a related post, but the additional ticks there mess up with the other ticks. I want to keep the rest of my major and minor ticks the intact.

Answer 1

llp = LogLogPlot[f[x], {x, 10^4, 10^7}, 
   Prolog -> {Line[{{Log[5*10^6], Log[f[5*10^3]]}, {Log[5*10^6], Log[f[5*10^6]]}}], 
     Line[{{Log[5*10^3], Log[f[5*10^6]]}, {Log[5*10^6], Log[f[5*10^6]]}}]}];

Extract the ticks used in llp using AbsoluteOptions:

{tx, ty} = Ticks /. AbsoluteOptions[llp, Ticks];

or generate them using the function Charting`ScaledTicks

{tx2, ty2} = Charting`ScaledTicks[{Log, Exp}][##, {6, 6}] & @@@ PlotRange[llp];

Specify the new ticks:

newxtick = {N@Log[5*10^6], Style[N@5*10^6, Red], {0.03, 0.01}, 
     {Red, AbsoluteThickness[3]}};

newytick = {N@Log[f[5*10^6]], Style[N@f[5*10^6], Red], {0.03, 0.01}, 
    {Red, AbsoluteThickness[3]}};

Add them to the original ticks list:

ticks = {DeleteDuplicatesBy[First]@Prepend[tx2, newxtick], 
   DeleteDuplicatesBy[First]@Prepend[ty2, newytick]};

Use Show with the option Ticks -> ticks:

Show[llp, Ticks -> ticks, ImageSize -> Large]

Answer 2

Try this:

f[x_] = 10^(-10)*Log[x];

Plot[f[x], {x, 10^4, 10^7}, ScalingFunctions -> {"Log", "Log"}, 
 Ticks -> {{5.*10^4, 5.*10^5, 5.*10^6} /. 
    x_?NumberQ -> {x, ScientificForm[x]}, {1.0*10^-9, 
    1.25*10^-9, {f[5*10^6] // N, 
     NumberForm[f[5*10^6] // N, {2, 1}]}}}, 
 Epilog -> {Line[{{Log[5*10^6], Log[f[5*10^3]]}, {Log[5*10^6], 
      Log[f[5*10^6]]}}], 
   Line[{{Log[5*10^3], Log[f[5*10^6]]}, {Log[5*10^6], 
      Log[f[5*10^6]]}}]}]

Have fun!