513. a) logs x=2; 6) logo, x= -1; B) logo x= -1; r) lg x=2. 514. a) log: (2x-4)=-2; 2 B) log0,3 (5+2x)=1; 6) log (x²+2x+3)=logx 6; r) log2 (3-x)=0. 515. a) (0,2)4-*=3; б) 5*²=7; в) 32-3x=8; г) 72х =4. Решите неравенства (516-517). 516. a) log3x2;6) logo.5x-2; в) log0,7 x <1; г) log2,5 x<2. 517. a) log4 (x-2)<2; B) logs (3x+1)>2; 6) log: (3-2x)> -1; r) log (4x+1)<-2. Решите уравнения (518-520). 3 7 518. a) loga x=2loga 3+loga 5; 6) 1g (x-9)+1g (2x-1)=2; B) loga x=loga 10-loga 2; г) log3 (x+1)+log3 (x+3)=1. 519. a) log2(x-4)+log2 (2x-1)=log2 3; 6) 1g (3x²+12x+19)-1g (3x+4)=1; B) 1g (x²+2x-7)-1g (x-1)=0; r) logs (x²+8)-logs (x+1)=3 logs 2. 520. a) logix+log4 √x-1,5=0; 6) 1g²x-lg x²+1=0; B) log x-log5 x=2; r) log3x-2 log3x-3=0. 521. Решите систему уравнений: a) (x+y=7, (1g x+1g y=1; в) (x+y=34, log2x+log2y=6; 6) (log4 (x+y)=2, ( log3x+loga y=2+log3 7; r) (log4 x-log4 y=0, (x²-5y²+4=0. Решите уравнения (522-524). 1 6 522. a) Igx+1+1gx+5=1; B) 1g (5x-4) 21g x = 1; 6) log2= 1 4 15 X log28 ; 1 r) Igx-6+ Igx+2=1; 5 523. a) loga x=log√2+log13; 6) logx 2-log4x+7=0; B) log3x-2 log, x=6; 3 r) log25 x+logs x=log, √8. 524. a) log2 (9-2*)=3-x; 6) log2 (25*+3-1)=2+log2 (5x+3+1);

1. 513. a) \( \log_5 x = 2 \)

Логика такая:

\[ x = 5^2 = 25 \]

Ответ: \( x = 25 \)

• б) \( \log_{0.4} x = -1 \)

\[ x = (0.4)^{-1} = \frac{1}{0.4} = \frac{10}{4} = \frac{5}{2} = 2.5 \]

Ответ: \( x = 2.5 \)

• в) \( \log_9 x = -\frac{1}{2} \)

\[ x = 9^{-\frac{1}{2}} = \frac{1}{\sqrt{9}} = \frac{1}{3} \]

Ответ: \( x = \frac{1}{3} \)

• г) \( \lg x = 2 \)

\[ x = 10^2 = 100 \]

Ответ: \( x = 100 \)

• 514. a) \( \log_{\frac{1}{2}} (2x - 4) = -2 \)

\[ 2x - 4 = (\frac{1}{2})^{-2} = 4 \]

\[ 2x = 8 \]

\[ x = 4 \]

Ответ: \( x = 4 \)

• б) \( \log_{\pi} (x^2 + 2x + 3) = \log_{\pi} 6 \)

\[ x^2 + 2x + 3 = 6 \]

\[ x^2 + 2x - 3 = 0 \]

\[ (x + 3)(x - 1) = 0 \]

\[ x = -3, x = 1 \]

Ответ: \( x = -3, x = 1 \)

• в) \( \log_{0.3} (5 + 2x) = 1 \)

\[ 5 + 2x = 0.3 \]

\[ 2x = -4.7 \]

\[ x = -2.35 \]

Ответ: \( x = -2.35 \)

• г) \( \log_2 (3 - x) = 0 \)

\[ 3 - x = 1 \]

\[ x = 2 \]

Ответ: \( x = 2 \)

• 515. a) \( (0.2)^{4 - x} = 3 \)

\[ (\frac{1}{5})^{4 - x} = 3 \]

\[ 5^{x - 4} = 3 \]

\[ x - 4 = \log_5 3 \]

\[ x = \log_5 3 + 4 \]

Ответ: \( x = \log_5 3 + 4 \)

• б) \( 5^{x^2} = 7 \)

\[ x^2 = \log_5 7 \]

\[ x = \pm \sqrt{\log_5 7} \]

Ответ: \( x = \pm \sqrt{\log_5 7} \)

• в) \( 3^{2 - 3x} = 8 \)

\[ 2 - 3x = \log_3 8 \]

\[ 3x = 2 - \log_3 8 \]

\[ x = \frac{2 - \log_3 8}{3} \]

Ответ: \( x = \frac{2 - \log_3 8}{3} \)

• г) \( 7^{2x} = 4 \)

\[ 2x = \log_7 4 \]

\[ x = \frac{\log_7 4}{2} \]

Ответ: \( x = \frac{\log_7 4}{2} \)

• 522. a)

\[ \frac{1}{\lg x + 1} + \frac{6}{\lg x + 5} = 1 \]

Пусть \( y = \lg x \), тогда:

\[ \frac{1}{y + 1} + \frac{6}{y + 5} = 1 \]

\[ \frac{y + 5 + 6(y + 1)}{(y + 1)(y + 5)} = 1 \]

\[ y + 5 + 6y + 6 = y^2 + 6y + 5 \]

\[ y^2 - y - 6 = 0 \]

\[ (y - 3)(y + 2) = 0 \]

\[ y = 3, y = -2 \]

Обратная замена:

\[ \lg x = 3 \Rightarrow x = 10^3 = 1000 \]

\[ \lg x = -2 \Rightarrow x = 10^{-2} = 0.01 \]

Ответ: \( x = 1000, x = 0.01 \)

• б)

\[ \log_2 \frac{x}{4} = \frac{15}{\log_2 \frac{x}{8}} \]

Пусть \( y = \log_2 x \), тогда:

\[ \log_2 \frac{x}{4} = \log_2 x - \log_2 4 = y - 2 \]

\[ \log_2 \frac{x}{8} = \log_2 x - \log_2 8 = y - 3 \]

\[ y - 2 = \frac{15}{y - 3} \]

\[ (y - 2)(y - 3) = 15 \]

\[ y^2 - 5y + 6 = 15 \]

\[ y^2 - 5y - 9 = 0 \]

\[ y = \frac{5 \pm \sqrt{25 + 36}}{2} = \frac{5 \pm \sqrt{61}}{2} \]

\[ x = 2^{\frac{5 \pm \sqrt{61}}{2}} \]

Ответ: \( x = 2^{\frac{5 \pm \sqrt{61}}{2}} \)