Answer:
The graph of the function [tex]f\left(x\right)=\:\log _{10}\:x-3[/tex] is attached below.
Step-by-step explanation:
Considering the function
[tex]f\left(x\right)=\:\log _{10}\:x-3[/tex]
[tex]\mathrm{Domain\:of\:}\:\log _{10}\left(x\right)-3\::\quad \begin{bmatrix}\mathrm{Solution:}\:&\:x>0\:\\ \:\mathrm{Interval\:Notation:}&\:\left(0,\:\infty \:\right)\end{bmatrix}[/tex]
[tex]\mathrm{Range\:of\:}\log _{10}\left(x\right)-3:\quad \begin{bmatrix}\mathrm{Solution:}\:&\:-\infty \:<f\left(x\right)<\infty \\ \:\mathrm{Interval\:Notation:}&\:\left(-\infty \:,\:\infty \:\right)\end{bmatrix}[/tex]
Determining x-intercept:
[tex]\mathrm{x-intercept\:is\:a\:point\:on\:the\:graph\:where\:}y=0[/tex]
[tex]\log _{10}\left(x\right)-3=0[/tex]
[tex]\log _{10}\left(x\right)=3[/tex]
Using the logarithmic definition
[tex]\mathrm{If}\:\log _a\left(b\right)=c\:\mathrm{then}\:b=a^c[/tex]
[tex]\log _{10}\left(x\right)=3\quad \Rightarrow \quad \:x=10^3[/tex]
[tex]x=1000[/tex]
so the x-intercept = (1000, 0)
Determining y-intercept:
[tex]y\mathrm{-intercept\:is\:the\:point\:on\:the\:graph\:where\:}x=0[/tex]
[tex]\mathrm{Since}\:x=0\:\mathrm{is\:not\:in\:domain}[/tex]
[tex]\mathrm{No\:y-axis\:interception\:point}[/tex]
Therefore, the graph of the function [tex]f\left(x\right)=\:\log _{10}\:x-3[/tex] is attached below.
