[tex]Domain:\\7x+3>0\qquad\text{subtract 3 from both sides}\\7x>-3\qquad\text{divide both sides by 7}\\x>-\dfrac{3}{7}[/tex]
[tex]x-intercept\ for\ y=0:\\\\\log(7x+3)-2=0\qquad\text{add 2 to both sides}\\\\\log(7x+3)=2\\\\\log(7x+3)=\log10^2\iff7x+3=100\qquad\text{subtract 3 from both sides}\\\\7x=97\qquad\text{divide both sides by 7}\\\\x\approx13.86\\\\y-intercept\ for\ x=0:\\\\y=\log(7(0)+3)-2\\\\y=\log(3)-2\approx-1.52\\\\Answer:\\x-intercept=\boxed{13.86}\\y-intercept=\boxed{-1.52}[/tex]
