Latihan Matematika Peminatan Kelas X Persamaan Logaritma
# 10
Pilgan

Hasil dari (3log⁡6)2−(3log⁡2)23log⁡12\frac{\left(^3\log6\right)^2-\left(^3\log2\right)^2}{^3\log\sqrt{12}} adalah ....

A

1

B

2

C

12\frac{1}{2}

D

12

E

14\frac{1}{4}

Pembahasan:

Gunakan sifat eksponen a2−b2=(a+b)(a−b)a^2-b^2=\left(a+b\right)\left(a-b\right)

⇔(3log⁡6)2−(3log⁡2)23log⁡12\Leftrightarrow\frac{\left(^3\log6\right)^2-\left(^3\log2\right)^2}{^3\log\sqrt{12}}

⇔(3log⁡6+3log⁡2)(3log⁡6−3log⁡2)3log⁡12\Leftrightarrow\frac{\left(^3\log6+^3\log2\right)\left(^3\log6-^3\log2\right)}{^3\log\sqrt{12}}

Gunakan sifat-sifat logaritma alog⁡b+alog⁡c=alog⁡b×c^a\log b+^a\log c=^a\log b\times c dan alog⁡b−alog⁡c=alog⁡ (bc)^a\log b-^a\log c=^a\log\ \left(\frac{b}{c}\right)

⇔(3log⁡6×2)(3log⁡ 62)3log⁡12\Leftrightarrow\frac{\left(^3\log6\times2\right)\left(^3\log\ \frac{6}{2}\right)}{^3\log\sqrt{12}}

⇔(3log⁡12)(3log⁡ 3)3log⁡12\Leftrightarrow\frac{\left(^3\log12\right)\left(^3\log\ 3\right)}{^3\log\sqrt{12}}

Gunakan sifat logaritma alog⁡a=1^a\log a=1

⇔(3log⁡12)×13log⁡12\Leftrightarrow\frac{\left(^3\log12\right)\times1}{^3\log\sqrt{12}}

Gunakan sifat logaritma alog⁡calog⁡b=blog⁡c\frac{^a\log c}{^a\log b}=^b\log c

⇔3log⁡123log⁡12\Leftrightarrow\frac{^3\log12}{^3\log\sqrt{12}}

⇔12log⁡12\Leftrightarrow^{\sqrt{12}}\log12

⇔  1212log⁡12\Leftrightarrow^{\ \ 12^{\frac{1}{2}}}\log12

Gunakan sifat logaritma anlog⁡am=mn×alog⁡a^{a^n}\log a^m=\frac{m}{n}\times^a\log a

⇔ 112×12log⁡12\Leftrightarrow\ \frac{1}{\frac{1}{2}}\times^{12}\log12

⇔ 2×1=2\Leftrightarrow\ 2\times1=2

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