Решите уравнения (512-515). Упражнения 512. - a) 9*=0,7; б) 0,3*=7; в) 2* = 10; г) 10*=л. 513.- a) logs x=2; 6) logo,4 x = -1; B) logg x=-1; r) lg x = 2. 2 514. - a) log₁ (2x - 4) = -2; 2 в) log0,3 (5+2x) = 1; 6) log (x²+2x+3)=log, 6; г) log2 (3 - x) = 0. 515.- a) 0,24-x=3; б) 5*² = 7; в) 32-3x = 8; г) 72х = 4.

Ответ: смотри решение ниже

512

1. a) \(9^x = 0{,}7\)
   • Логарифмируем обе части по основанию 10: \(\lg(9^x) = \lg(0{,}7)\)
   • \(x \cdot \lg(9) = \lg(0{,}7)\)
   • \(x = \frac{\lg(0{,}7)}{\lg(9)} \approx -0{,}166\)
2. б) \(0{,}3^x = 7\)
   • \(\lg(0{,}3^x) = \lg(7)\)
   • \(x \cdot \lg(0{,}3) = \lg(7)\)
   • \(x = \frac{\lg(7)}{\lg(0{,}3)} \approx -1{,}771\)
3. в) \(2^x = 10\)
   • \(\lg(2^x) = \lg(10)\)
   • \(x \cdot \lg(2) = 1\)
   • \(x = \frac{1}{\lg(2)} \approx 3{,}322\)
4. г) \(10^x = \pi\)
   • \(\lg(10^x) = \lg(\pi)\)
   • \(x = \lg(\pi) \approx 0{,}497\)

513

1. a) \(\log_5 x = 2\)
   • \(x = 5^2\)
   • \(x = 25\)
2. б) \(\log_{0{,}4} x = -1\)
   • \(x = (0{,}4)^{-1}\)
   • \(x = \frac{1}{0{,}4} = 2{,}5\)
3. в) \(\log_9 x = -\frac{1}{2}\)
   • \(x = 9^{-\frac{1}{2}}\)
   • \(x = \frac{1}{\sqrt{9}} = \frac{1}{3}\)
4. г) \(\lg x = 2\)
   • \(x = 10^2\)
   • \(x = 100\)

514

1. a) \(\log_{\frac{1}{2}} (2x - 4) = -2\)
   • \(2x - 4 = (\frac{1}{2})^{-2}\)
   • \(2x - 4 = 4\)
   • \(2x = 8\)
   • \(x = 4\)
2. б) \(\log_{\pi} (x^2 + 2x + 3) = \log_{\pi} 6\)
   • \(x^2 + 2x + 3 = 6\)
   • \(x^2 + 2x - 3 = 0\)
   • \((x + 3)(x - 1) = 0\)
   • \(x = -3\) или \(x = 1\)
3. в) \(\log_{0{,}3} (5 + 2x) = 1\)
   • \(5 + 2x = 0{,}3^1\)
   • \(5 + 2x = 0{,}3\)
   • \(2x = -4{,}7\)
   • \(x = -2{,}35\)
4. г) \(\log_2 (3 - x) = 0\)
   • \(3 - x = 2^0\)
   • \(3 - x = 1\)
   • \(x = 2\)

515

1. a) \(0{,}2^{4-x} = 3\)
   • \(\lg(0{,}2^{4-x}) = \lg(3)\)
   • \((4-x) \cdot \lg(0{,}2) = \lg(3)\)
   • \(4-x = \frac{\lg(3)}{\lg(0{,}2)}\)
   • \(4-x = \frac{\lg(3)}{\lg(\frac{1}{5})}\)
   • \(x = 4 - \frac{\lg(3)}{\lg(\frac{1}{5})} \approx 4 - (-0{,}683) \approx 4{,}683\)
2. б) \(5^{x^2} = 7\)
   • \(\lg(5^{x^2}) = \lg(7)\)
   • \(x^2 \cdot \lg(5) = \lg(7)\)
   • \(x^2 = \frac{\lg(7)}{\lg(5)}\)
   • \(x = \pm \sqrt{\frac{\lg(7)}{\lg(5)}} \approx \pm 1{,}209\)
3. в) \(3^{2-3x} = 8\)
   • \(\lg(3^{2-3x}) = \lg(8)\)
   • \((2-3x) \cdot \lg(3) = \lg(8)\)
   • \(2-3x = \frac{\lg(8)}{\lg(3)}\)
   • \(3x = 2 - \frac{\lg(8)}{\lg(3)}\)
   • \(x = \frac{1}{3} \cdot (2 - \frac{\lg(8)}{\lg(3)}) \approx -0{,}295\)
4. г) \(7^{2x} = 4\)
   • \(\lg(7^{2x}) = \lg(4)\)
   • \(2x \cdot \lg(7) = \lg(4)\)
   • \(2x = \frac{\lg(4)}{\lg(7)}\)
   • \(x = \frac{1}{2} \cdot \frac{\lg(4)}{\lg(7)} \approx 0{,}356\)

Ответ: смотри решение выше