logticks[a_,b_]:=First/@Charting`ScaledTicks[{Log,Exp}][a,b]//Exp;
MapThread[Themes`AddThemeRules["LogGrid",#,GridLines->#2,Frame->True]&,{{LogPlot|ListLogPlot,LogLinearPlot|ListLogLinearPlot,LogLogPlot|ListLogLogPlot},{{Automatic,logticks},{logticks,Automatic},logticks}}];

Plot[4^x,{x,1,5},GridLinesStyle->LightGray,PlotStyle->{
{Thickness[0.005],Blue}},
PlotRange->{{1.,5},{10.,1000}},
FrameLabel->{
{Style["y",24,Black,FontFamily->"Helvetica"], None},
{Style["x ",24,FontFamily->"Helvetica"], None}
},
GridLinesStyle->LightGray,PlotTheme->"Grid",Frame->True,
FrameTicksStyle->Directive[Black, 24,FontFamily->"Helvetica"],
Frame->True,
Background -> White,
AspectRatio->0.8,
ImageSize->700,
LabelStyle->Directive[20,FontFamily->"Helvetica"],
PlotLabel->Style["Linear Plot, y = 4^x ",32,FontFamily->"Helvetica"]]

LogPlot[4^x,{x,1,5},PlotStyle->{
{Thickness[0.005],Blue}},
PlotRange->{{1.,5},{1.,1000}},
FrameLabel->{
{Style["y",24,Black,FontFamily->"Helvetica"], None},
{Style["x ",24,FontFamily->"Helvetica"], None}
},
GridLinesStyle->LightGray,PlotTheme->"LogGrid",Frame->True,
FrameTicksStyle->Directive[Black, 24,FontFamily->"Helvetica"],
Frame->True,
Background -> White,
AspectRatio->0.8,
ImageSize->700,
LabelStyle->Directive[20,FontFamily->"Helvetica"],
PlotLabel->Style["Log-linear Plot, y = 4^x",32,FontFamily->"Helvetica"]]

({{}, {\[Placeholder]}})