Exercise Zone : Pertidaksamaan Logaritma

Berikut ini adalah kumpulan soal mengenai pertidaksamaan logaritma tipe standar. Jika ada jawaban yang salah, mohon dikoreksi melalui komentar. Jika ingin bertanya soal, silahkan tulis di halaman Tanya Soal, Grup Facebook, atau Grup Telegram. Terima kasih.

No. 1

Himpunan penyelesaian pertidaksamaan {^{\frac13}\negmedspace\log(2x-1)+{^{\frac13}\negmedspace\log}(2-x)\geq2\cdot{^{\frac13}\negmedspace\log}x} adalah

1. {\left\{x\in R\ \middle\vert\ \dfrac23\leq x\leq1\right\}}
2. {\left\{x\in R\ \middle\vert\ x\leq\dfrac23\text{ atau }x\geq1\right\}}
3. {\left\{x\in R\ \middle\vert\ \dfrac12\leq x\leq\dfrac23\text{ atau }1\leq x\leq2\right\}}

4. {\left\{x\in R\ \middle\vert\ \dfrac12\lt x\leq\dfrac23\text{ atau }1\leq x\lt2\right\}}
5. {\left\{x\in R\ \middle\vert\ x\leq\dfrac12\text{ atau }x\gt2\right\}}

\(\begin{aligned} ^{\frac13}\negmedspace\log(2x-1)+{^{\frac13}\negmedspace\log}(2-x)&\geq2\cdot{^{\frac13}\negmedspace\log}x\\ ^{\frac13}\negmedspace\log\left((2x-1)(2-x)\right)&\geq{^{\frac13}\negmedspace\log}x^2\\ ^{\frac13}\negmedspace\log\left(4x-2x^2-2+x\right)&\geq{^{\frac13}\negmedspace\log}x^2\\ ^{\frac13}\negmedspace\log\left(-2x^2+5x-2\right)&\geq{^{\frac13}\negmedspace\log}x^2\\ -2x^2+5x-2&\leq x^2\\ -3x^2+5x-2&\leq0\\ 3x^2-5x+2&\geq0\\ (3x-2)(x-1)&\geq0 \end{aligned}\)
x\leq\dfrac23 atau x\geq1

Syarat:

\(\begin{aligned} 2x-1&\gt0\\ 2x&\gt1\\ x&\gt\dfrac12 \end{aligned}\)

\(\begin{aligned} 2-x&\gt0\\ -x&\gt-2\\ x&\lt2 \end{aligned}\)

\dfrac12\lt x\leq\dfrac23 atau \leq x\lt2

Jadi, Hp = \left\{x\in R\ \middle\vert\ \dfrac12\lt x\leq\dfrac23\text{ atau }1\leq x\lt2\right\}.
JAWAB: D