Question

Решить уравнение на картинке

Answer 1

6^log (6) x) ^log (6) x+x^log (6) x=12

x^log (6) x+x^log (6) x=12

2x^log (6) x=12

x^log (6) x=6

Прологарифмируем обе части по основанию

log (6) x^log (6) x-log (6) 6

log (6) x*log (6) x=1

log² (6) x=1

log (6) x=1⇒x=6

log (6) x=-1⇒x=1/6

Answer 2

ОДЗ : x > 0

$6^{log_{6}^{2} x } +x^{log_{6}x }=12\\\\(6^{log_{6} x })^{log_{6}x } +x^{log_{6}x }=12\\\\x^{log_{6} x}+x^{log_{6} x}=12\\\\2x^{log_{6} x}=12\\\\x^{log_{6} x}=6\\\\log_{6}x^{log_{6} x}=log_{6} 6\\\\log_{6} x*log_{6} x=1\\\\log_{6}^{2} x=1\\\\\left[\begin{array}{ccc}log_{6}x=-1 \\log_{6}x=1 \end{array}\right\\\\\\\left[\begin{array}{ccc}x_{1}=\frac{1}{6} \\x_{2}=6 \end{array}\right\\\\Otvet:\boxed{\frac{1}{6} \ ; \ 6}$