Question

упростить логарифмы. даю много

Answer 1

1)

$displaystyle sqrt[4]{frac{a^{log_ab+3}*b^{log_ba+3}}{a^{log_ab-1}*b^{log_ba-1}}}= sqrt[4]{frac{a^{log_ab}*a^3*b^{log_ba}*b^3}{a^{log_ab}/a * b^{log_ba}/b}}=\\= sqrt[4]{frac{b*a^3*a*b^3}{frac{b*a}{a*b}}}= sqrt[4]{a^4*b^4}=a*b$

2)

$displaystyle ((log^2_ab-log^2_ac)-(log_ab+log_ac))*frac{1}{log_ab/c}=\\=((log_ab-log_ac)(log_ab+log_ac)-log_abc)* frac{1}{log_ab/c}=\\= frac{log_ab/c*log_abc-log_abc}{log_ab/c}= frac{log_abc(log_ab/c-1)}{log_ab/c}$

можно и дальше преобразовать

$displaystyle =log_abc-frac{log_abc}{log_ab/c}=log_abc-log_abc*log_{b/c}a=log_abc(1-log_{b/c}a)$