043.13. a) log₄ 4 ⋅ log₃ 9 : log₄(1/4); б) log√3 3√3 : log₁/7 √49 ⋅ log₅ √5; в) log₃ 81 : log₀.₅ 2 ⋅ log₅ 125; г) log₅√5 ⋅ log₀.₃ 0,3 : lg 10√0,1.

Решение:

а)

• \[ \log_4 4 = 1 \]
• \[ \log_3 9 = \log_3 3^2 = 2 \]
• \[ \log_4 \frac{1}{4} = \log_4 4^{-1} = -1 \]
• \[ 1 \cdot 2 : (-1) = 2 : (-1) = -2 \]

б)

• \[ \log_{\sqrt{3}} 3\sqrt{3} = \log_{3^{1/2}} 3^{3/2} = \frac{3/2}{1/2} = 3 \]
• \[ \log_{1/7} \sqrt{49} = \log_{7^{-1}} 7 = \frac{1}{-1} = -1 \]
• \[ \log_5 \sqrt{5} = \log_5 5^{1/2} = \frac{1}{2} \]
• \[ 3 : (-1) \cdot \frac{1}{2} = -3 \cdot \frac{1}{2} = -\frac{3}{2} \]

в)

• \[ \log_3 81 = \log_3 3^4 = 4 \]
• \[ \log_{0,5} 2 = \log_{1/2} 2 = \log_{2^{-1}} 2 = \frac{1}{-1} = -1 \]
• \[ \log_5 125 = \log_5 5^3 = 3 \]
• \[ 4 : (-1) \cdot 3 = -4 \cdot 3 = -12 \]

г)

• \[ \log_5 \sqrt{5} = \frac{1}{2} \]
• \[ \log_{0,3} 0,3 = 1 \]
• \[ \lg 10\sqrt{0,1} = \lg 10 \cdot \sqrt{10^{-1}} = \lg 10 \cdot 10^{-1/2} = \lg 10^{1 - 1/2} = \lg 10^{1/2} = \frac{1}{2} \]
• \[ \frac{1}{2} \cdot 1 : \frac{1}{2} = \frac{1}{2} : \frac{1}{2} = 1 \]

Ответ: а) -2; б) -3/2; в) -12; г) 1.