4.6.4 Indeks, Surd dan Logaritma, SPM Praktis (Soalan Pendek)

Soalan 12

Selesaikan persamaan,

\log_{2} 5 \sqrt{x} + \log_{4} 16 x = 6

Penyelesaian:

\begin{array}{l} \log_{2} 5 \sqrt{x} + \log_{4} 16 x = 6 \\ \log_{2} 5 \sqrt{x} + \frac{\log_{2} 16 x}{\log_{2} 4} = 6 \\ \log_{2} 5 \sqrt{x} + \frac{\log_{2} 16 x}{2} = 6 \\ 2 \log_{2} 5 \sqrt{x} + \log_{2} 16 x = 12 \\ \log_{2} {(5 \sqrt{x})}^{2} + \log_{2} 16 x = 12 \\ \log_{2} (25 x) + \log_{2} 16 x = 12 \\ \log_{2} (25 x) (16 x) = 12 \\ \log_{2} 400 x^{2} = 12 \\ 400 x^{2} = 2^{12} \\ x^{2} = 10.24 \\ x = 3.2 \end{array}

Soalan 13

Diberi bahawa 2 log₂ (x – y) = 3 + log₂ x + log₂y. Buktikan x² + y²– 10xy = 0.

Penyelesaian:

2 log₂ (x – y) = 3 + log₂x + log₂ y

log₂ (x– y)² = log₂ 8 + log₂ x + log₂y

log₂ (x– y)² = log₂ 8xy

(x – y)² = 8xy

x²– 2xy + y² = 8xy

x² + y² – 10xy = 0 (terbukti)

Soalan 14

Diberi bahawa 2 log₂ (x + y) = 3 + log₂ x + log₂y. Buktikan x² + y²= 6xy.

Penyelesaian:

2 log₂ (x + y) = 3 + log₂x + log₂ y

log₂ (x+ y)² = log₂ 8 + log₂ x + log₂y

log₂ (x+ y)² = log₂ 8xy

(x + y)² = 8xy

x²+ 2xy + y² = 8xy

x² + y² = 6xy (terbukti)