The domain and range of the logarithmic function are
[tex]\begin{gathered} \text{domain}(\log x)=(0,\infty) \\ \text{range}(\log x)=(-\infty,\infty) \end{gathered}[/tex]
Therefore, if
[tex]g(x)=\log (x-4)-8[/tex]
We require that
[tex]\begin{gathered} x-4>0 \\ \Rightarrow x>4 \end{gathered}[/tex]
Notice that the -8 term does not affect the range of function g(x); thus,
[tex]\begin{gathered} \text{domain}(g(x))=(4,\infty) \\ \text{range}(g(x))=(-\infty,\infty) \end{gathered}[/tex]
Set g(x)=-8; then,
[tex]\begin{gathered} \Rightarrow\log (x-4)-8=-8 \\ \Rightarrow\log (x-4)=0 \\ \Rightarrow x=5 \end{gathered}[/tex]
Therefore, y=-8 is not an asymptote of g(x), and, as shown above, the domain and range of g(x) are x>4, y->all real numbers.
Calculate the limit when x->4 as shown below,
[tex]\lim _{x\to4}g(x)=(\lim _{x\to4}\log (x-4))-8=(-\infty)-8=-\infty[/tex]
Therefore, there is a vertical asymptote at x=4
