Graph and Domain of Logarithmic functions

Plot[
{Log[x],Log10[x]},
{x,-1,10},
PlotLegends->"Expressions"
]

Domain of both is [0,+∞],
Range is [-∞,?]

What is the domain of ln(x+3)? -log(x-2)?

We say that there is a vertical asymptote of log[x] at x=0, because (lim_(x→0+ ))log[x]=-∞.  
That is, the graph approaches the vertical line x=0.

Graph and Domain of exponential functions

Plot[
{E^x,10^x},
{x,-1,1},
PlotLegends->"Expressions"
]

Domain of both is [-∞,+∞],
Range is [0,+∞]

We say that there is a horizontal asymptote of (e^x) at y=0, because (lim_(x→-∞ ))(e^x)=0.  
That is, the graph approaches the horizontal line y=0.

If k>0, then (lim_(x->+∞))(e^kx)=+∞, and (lim_(x->-∞))(e^kx)=0.

Exponential and Log functions are inverse functions of each other

Plot[
{Exp[x],Log[x],   x},
{x,-1,5},PlotRange->{-1,5}, AspectRatio->1
]

Plot[
{10^x,Log10[x],   x},
{x,-1,5},PlotRange->{-1,5}, AspectRatio->1
]

Vertical and horizontal asymptotes

({{f(x)=({{(1/((x-1)(x+2)))+3}})}})

Is there any x value for which this “blows up”?

What is the limit of f(x) as as x→+∞? as x→-∞?  [You can always get close, by calculating for very large positive, or very large negative values.  For example, f(1000000) and f(-1000000).]

f[x_]:= (1/((x-1)(x+2)))+3;
{f[1000000.],f[-1000000.]}

Plot[
{f[x],3},
{x,-7,7},
PlotRange->{-4,8}
 ]

Will this function have any horizontal asymptotes?

({{({{(20/(5+(e^(0.2x))))}})}})

Try to calculate the limits:  as x→+∞; as x→-∞.

If k>0, then (lim_(x->+∞))(e^kx)=+∞, and (lim_(x->-∞))(e^kx)=0.

g[x_]:=({{(20/(5+(E^(0.2x))))}});
{g[10000.],g[-10000.]}
Plot[({{(20/(5+(E^(0.2x))))}}),{x,-30,40},PlotRange->{-2.2,6.2}]