Question

Jika [tex]\displaystyle ^a\log b+^b\log a=3[/tex] tentukan nilai dari [tex]\displaystyle \sqrt{\frac{\left ( ^a\log^3 b+^a\log^2 b \right )\left ( ^a\log^5 b+1 \right )}{^a\log^5 b}}[/tex]

Answer 1

[tex]^alogb+^bloga=3\\\\^alogb+\frac{1}{^alogb} =3\\\\misal : x = ^alogb,maka :\\\\x+\frac{1}{x}=3[/tex]

[tex]x+\frac{1}{x}=3\\\\(x+\frac{1}{x})^2=3^2\\\\x^2+2+\frac{1}{x^2}=9\\\\x^2+\frac{1}{x^2}=7\\\\\\\\x+\frac{1}{x}=3\\\\(x+\frac{1}{x})^3=3^3\\\\x^3+3x+\frac{3}{x} +\frac{1}{x^3}=27\\\\x^3+3(x+\frac{1}{x}) +\frac{1}{x^3}=27\\\\x^3+3(3) +\frac{1}{x^3}=27\\\\x^3+9 +\frac{1}{x^3}=27\\\\x^3+\frac{1}{x^3}=18\\[/tex]

[tex]\sqrt{\frac{(^alog^3b+^alog^2b)(^alog^5b+1)}{^alog^5b} } \\\\=\sqrt{\frac{(x^3+x^2)(x^5+1)}{x^5} } \\\\=\sqrt{\frac{x^8+x^3+x^7+x^2}{x^5} } \\\\=\sqrt{x^3+\frac{1}{x^2}+x^2+\frac{1}{x^3}} \\\\=\sqrt{x^3+\frac{1}{x^3}+x^2+\frac{1}{x^2}} \\\\=\sqrt{18+7} \\\\=\sqrt{25}\\\\=5[/tex]