ЛОГАРИФМЫ
а) ㏒²₅Х + ㏒₅Х = 2
б) ㏒₂(Х+1) = 3-㏒₂(х+3)
в) ㏑(х-2) - 1/2㏑(3х-6)=㏑2

1)
[tex]\log^2_5x + \log_5x = 2\\ \log_5x = t\\\\ x \ \textgreater \ 0\\\\ t^2+t-2=0\\ t_1 = 1, t_2 = -2\\\\ \log_5x=1\\ x_1 = 5\\\\ \log_5x = -2\\ x_2 = \frac{1}{25}\\[/tex]

2)
[tex]\log_2(x+1) = 3 - \log_2(x+3)\\\\ x + 1 \ \textgreater \ 0\\ x+3 \ \textgreater \ 0\\ x \ \textgreater \ -1\\\\ \log_2(x+1) +\log_2(x+3)=3\\ (x+1)(x+3) = 8\\ x^2 + 3x + x + 3-8=0\\ x^2+4x-5=0\\ x=1\\[/tex]

3)
[tex]\ln(x-2) - \frac{1}{2}\ln(3x-6) = \ln2\\\\ x - 2 \ \textgreater \ 0\\ 3x-6 \ \textgreater \ 0\\ x \ \textgreater \ 2\\\\ \ln(x-2) - \ln(\sqrt{3x-6}) = \ln2\\\\ \ln\frac{x-2}{\sqrt{3x-6}} = \ln2\\\\ \frac{x-2}{\sqrt{3x-6}} = 2\\ x - 2 = 2\sqrt{3x-6}\\ (x-2)^2 = 4(3x-6)\\ x^2-4x+4-12x+24=0\\ x^2-16x+28=0\\ D = 64-28 = 36\\ x_1 = 8-6=2, \varnothing\\ x_2=8+6=14\\[/tex]