\n$g(x)=\\log x-\\log (x-2)$<\/p>\n<\/blockquote>\n
\\[\\begin{array}{*{35}{l}}
\n{{D}_{f}} & = & \\left\\{ x\\in \\mathbb{R}:\\frac{x}{x-2}>0 \\right\\}\u00a0 \\\\
\n{} & = & \\left\\{ x\\in \\mathbb{R}:(x>0\\wedge x-2>0)\\vee (x<0\\wedge x-2<0) \\right\\}\u00a0 \\\\
\n{} & = & \\left\\{ x\\in \\mathbb{R}:(x>0\\wedge x>2)\\vee (x<0\\wedge x<2) \\right\\}\u00a0 \\\\
\n{} & = & \\left\\{ x\\in \\mathbb{R}:x>2\\vee x<0 \\right\\}\u00a0 \\\\
\n{} & = & \\left] -\\infty ,0 \\right[\\cup \\left] 2,+\\infty\u00a0 \\right[\u00a0 \\\\
\n\\end{array}\\]<\/p>\n
\\[\\begin{array}{*{35}{l}}
\n{{D}_{g}} & = & \\left\\{ x\\in \\mathbb{R}:x>0\\wedge x-2>0 \\right\\}\u00a0 \\\\
\n{} & = & \\left\\{ x\\in \\mathbb{R}:x>0\\wedge x>2 \\right\\}\u00a0 \\\\
\n{} & = & \\left] 2,+\\infty\u00a0 \\right[\u00a0 \\\\
\n\\end{array}\\]<\/p>\n
Como ${{D}_{f}}\\ne {{D}_{g}}$, ent\u00e3o as fun\u00e7\u00f5es $f$ e $g$ n\u00e3o s\u00e3o id\u00eanticas.<\/p>\n
Contudo, como \\[\\log \\left( \\frac{x}{x-2} \\right)=\\log x-\\log (x-2)\\,,\\,\\forall x\\in \\left] 2,+\\infty\u00a0 \\right[\\] as fun\u00e7\u00f5es s\u00e3o id\u00eanticas quando se considera $f$\u00a0restrita ao dom\u00ednio de $g$.<\/p>\n