begin{tikzpicture}
begin{axis}[
  axis lines=middle,
  samples=100,
  ymax=5,legend pos=north east,
  legend style={draw=none}
]
addplot[forget plot,cyan,domain=0.001:8] {abs(log10(x))};
addplot[forget plot,red!70!black,domain=-2:4] {2+exp(abs(x))};
addplot[green,thick,domain=-2:3] {exp(x)};
addlegendentry{$e^{x}$};
node[pin={90:$f(x)=lvertlog xrvert$},inner sep=0pt]
  at (axis cs:{2,log10(2)}) {};
node[pin={0:$f(x)=e^{x}$},inner sep=0pt]
  at (axis cs:{1,exp(1)}) {};
node[anchor=north west]
  at (axis description cs:0,0.5)
  {$lvert e^{x}rvert$};
end{axis}
end{tikzpicture}

begin{tikzpicture}
begin{axis}[
  axis lines=middle,
  samples=100,
  ymax=5,legend pos=north east,
  legend style={draw=none}
]
addplot[forget plot,cyan,domain=0.001:8] {sin(x)};
addplot[forget plot,red!70!black,domain=-2:4] {2+exp(abs(x))};
addplot[green,thick,domain=-2:3] {exp(x)};
addlegendentry{$e^{x}$};
node[pin={90:$f(x)=lvertlog xrvert$},inner sep=0pt]
  at (axis cs:{2,log10(2)}) {};
node[pin={0:$f(x)=e^{x}$},inner sep=0pt]
  at (axis cs:{1,exp(1)}) {};
node[anchor=north west]
  at (axis description cs:0,0.5)
  {$lvert e^{x}rvert$};
end{axis}
end{tikzpicture}