\nEnunciado<\/b><\/span><\/p>\n
Seja $f$ a fun\u00e7\u00e3o definida em ${{\\mathbb{R}}^{+}}$ por $$f(x)={{\\log }_{4}}\\left( \\frac{{{x}^{2}}}{16} \\right)-{{\\log }_{4}}x$$<\/p>\n
- \n
- Mostre que $f(x)=-2+{{\\log }_{4}}x$, para qualquer $x\\in {{\\mathbb{R}}^{+}}$.<\/li>\n
- Determine a abcissa do ponto de interse\u00e7\u00e3o do gr\u00e1fico de $f$ com a reta de equa\u00e7\u00e3o $y=3$.<\/li>\n<\/ol>\nResolu\u00e7\u00e3o >><\/a><\/span><\/div><\/div>\n\n\nResolu\u00e7\u00e3o<\/b><\/span><\/p>\n
\n
Seja $f$ a fun\u00e7\u00e3o definida em ${{\\mathbb{R}}^{+}}$ por $$f(x)={{\\log }_{4}}\\left( \\frac{{{x}^{2}}}{16} \\right)-{{\\log }_{4}}x$$<\/p>\n<\/blockquote>\n
\u00ad<\/p>\n
- \n
- O dom\u00ednio da fun\u00e7\u00e3o \u00e9: $$\\begin{array}{*{35}{l}}
\n{{D}_{f}} & = & \\left\\{ x\\in \\mathbb{R}:\\left( \\frac{{{x}^{2}}}{16} \\right)>0\\wedge x>0 \\right\\}\u00a0 \\\\
\n{} & = & \\left\\{ x\\in \\mathbb{R}:x\\ne 0\\wedge x>0 \\right\\}\u00a0 \\\\
\n{} & = & {{\\mathbb{R}}^{+}}\u00a0 \\\\
\n\\end{array}$$<\/p>\nOra, $$\\begin{array}{*{35}{l}}
\nf(x) & = & {{\\log }_{4}}\\left( \\frac{{{x}^{2}}}{16} \\right)-{{\\log }_{4}}x\u00a0 \\\\
\n{} & = & {{\\log }_{4}}{{x}^{2}}-\\log {}_{4}16-{{\\log }_{4}}x\u00a0 \\\\
\n{} & = & 2{{\\log }_{4}}x-2-{{\\log }_{4}}x\u00a0 \\\\
\n{} & = & -2+{{\\log }_{4}}x\u00a0 \\\\
\n\\end{array}$$
\nLogo, $f(x)=-2+{{\\log }_{4}}x\\,\\,,\\forall x\\in {{\\mathbb{R}}^{+}}$.
\n\u00ad<\/p>\n<\/li>\n - Pretende-se determinar a solu\u00e7\u00e3o da equa\u00e7\u00e3o: \\[\\begin{array}{*{35}{l}}
\nf(x)=3 & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n-2+{{\\log }_{4}}x=3 & \\wedge\u00a0 & x\\in {{\\mathbb{R}}^{+}}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\n{{\\log }_{4}}x=5 & \\wedge\u00a0 & x\\in {{\\mathbb{R}}^{+}}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & \\begin{array}{*{35}{l}}
\nx={{4}^{5}} & \\wedge\u00a0 & x\\in {{\\mathbb{R}}^{+}}\u00a0 \\\\
\n\\end{array}\u00a0 \\\\
\n{} & \\Leftrightarrow\u00a0 & x=1024\u00a0 \\\\
\n\\end{array}\\]
\nPortanto, \u00e9 $1024$ a abcissa do ponto de interse\u00e7\u00e3o do gr\u00e1fico de $f$ com a reta de equa\u00e7\u00e3o $y=3$.![\"\"](\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/01\/12p203-8b.gif\" "\\"Solu\u00e7\u00e3o\\"")
<\/a>![\"\"](\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/01\/12p203-8a.gif\" "\\"Gr\u00e1fico\\"")
<\/a><\/li>\n<\/ol>\n
- O dom\u00ednio da fun\u00e7\u00e3o \u00e9: $$\\begin{array}{*{35}{l}}